Yes, their modems. On the connection that they provide for you.
A cable modem is considered CPE (customer premise equipment), meaning it is part of the infrastructure a telco uses to provide you with connectivity. Usually they own it, but in any case they have full control over it, as they should - it's part of their network. They may choose to delegate some configuration via a web GUI, but that's at their discretion -it's theirs to administer.
Business telecom has a formalized notion of a demarc (demarcation point), the place where the telco network ends and yours begins. AT&T owns and is responsible for the fiber/T1/POTS lines as they come through the wall, as well as the CPE (often a large rackmount Cisco router) to which it connects. Their contract is to provide connectivity on specific ethernet ports/fibre GBICs/whatever of that CPE. Whatever happens downstream of those ports is your problem, and whatever happens upstream is their problem.
Both sides will treat this connection as hostile - you'll have your own NATing router up and the telco's router, if it even has a configuration interface listening on your NIC, won't let you in. It would be inappropriate for AT&T to have any sort of access to the router you own and inappropriate for you to attempt any sort of access to AT&T's CPE.
Time Warner has been shifting recently towards placing WiFi on their side of the (logical) demarc. Which makes sense, since most people would rather not be responsible for administering any infrastructure - they just want Time Warner to deliver them WiFi. It sounds like you have this kind of setup, in which case Time Warner's access is not "backdoor" but "building owner" - you're renting a room.
If you'd prefer, you can (have them) turn off their WiFi, go buy a nice wireless router, and connect it to the modem. In this case Time Warner is providing you with a connection on an ethernet port; the device you've plugged in is your own (your side of the demarc) and they have no right to touch its configuration, nor are they responsible for it working correctly.
EDIT: The obvious analogy that would have simplified much of this is that a cable modem is like an electrical meter.
One of my cable installers told me that rate limiting is done in the cable modem, so people would run pirate firmware that eliminated the artifical limits and run at the natural limit of the connection. People had fun with this for a while until the network engineers figured it out, and now people exceeding the speed limit get their connections shut down pretty quickly. But anyway, it makes sense the the cable modem really isn't the customer's to control.
I own and control my own wireless router because I want to play with things like DD-WRT, use OpenDNS, etc., but I see the cable modem as no different from the utility box down the street.
With cable modems, does/can the provider push firmware?
The last I heard about it the different levels of service (bronze/silver/gold they were at the time, 5/10/20Mbit/s) are just based on the MAC the modem sends on this initial config/handshake. When I moved from 20 to 50 I was told to reboot the modem and it came up will an all new shiny more craptastic than ever web interface as well as setting it's WAN port to 50Mbit/s
The provider is able to change the settings through two primary methods.
1. DOCSIS configuration file - this is the file your modem downloads when it comes online and includes settings like your speeds (Upstream/Downstream service flows) and it also includes the SNMP settings (used for #2).
2. SNMP - The MSO can also remotely monitor and change your modem via SNMP. There is a large number of DOCSIS MIBs that every cable modem must support in order to get certified and there are also vendor specific MIBs that a modems manufacture will add to support specific features of that modem.
Without SNMP it would be very difficult to maintain a cable network. Other types of access networks have similar features.
I don't know why exactly it worked out that way, but the telephone system is designed to deal with consumer electronics and the coax cable system is not.
It didn't used to be - until the federal government busted AT&T's monopoly, it was illegal to connect non-AT&T equipment to the AT&T network, at least electrically. This is why in old movies you see people dialing ISPs by hand with a phone and then setting the handset down on a cradle - the cradle was the modem, and it interfaced with the telephone network acoustically. The demarc (at least as we know it today) was presumably introduced after the government told AT&T they had to let people have their own networks, rather than considering everything up through the phones to be part of the telco network.
The demarc for your DSL is the modem. If you provide your own modem you still are subject to AT&T's rules. You cannot legally connect to sensitive national infrastructure (like the phone network) without the provider being responsible for your access.
As far as connecting equipment, that significantly predates divestiture. Most people point to Carterfone as the salient decision that changed this: http://en.wikipedia.org/wiki/Carterfone
In reality, there was actually a prior decision which I can't seem to find right now where AT&T refused to allow people to put plastic privacy guards over the mouth piece on the grounds that it would damage the phone network. Ludicrous, but sensible from an economic standpoint.
EDIT: Found it, 1956 Hush-a-phone: http://en.wikipedia.org/wiki/Hush-A-Phone_v._United_States
In short, Telecom is complicated, the demarc is important and you don't own any part of the network from the modem on up.
Source: I am a telecom junkie and I work at 2600hz, the open source telecom company.
Lately cable companies have been pushing these hybrid modem/router combinations with things like wifi support built in. From a consumer standpoint, this seems very convenient, but the cable companies do this because it makes it easier for then. If you call up with an issue with "your Internet," they can remotely diagnose it and reset your modem/router. Make no mistake; they have more control over these than you do.
If you want to stop your ISP from having administrative access to your "private connection" (I assume you mean your wifi), then don't put the modem and the router in the same box. There's no other way around this.
You might be surprised at how much a provider support rep can see with this. Here's two screenshots from Cisco's product to give you a idea: http://images.newsfactor.com/images/super/larger-12-ClearAcc... http://cdn-static.zdnet.com/i/story/60/01/072589/clearaccess...
Moral of the story: Use the most basic cable/dsl modem that they'll give you, make it as close to bridge mode as you can, and use your own router.
Over here when you are BT's FTTC setup through any ISP the vDSL modem that hangs off your master socket (which can do more but is used in this arrangement to simple pick up the connection from the phone line and provide PPPoE on the ethernet port) is very definitely BT/OR's: they tell you not to mess with it, people who want to mess with it have to use hacks to get access to the UI (which is otherwise locked off), and if you plug something else in at that point you are officially not supported. If the router that you plug in to that came from your ISP then that is their's (usually you have to return it when you leave).
If you buy your own router (or "make" your own, people who have a small Linux machine on 24/7 for various things just set that up to talk PPPoE directly and skip the router altogether, neatly avoiding the limits of many "consumer grade" units (shoddy IPv6 support for instance) without shelling out for a much better device) only then do you truly have control of security at that point in the topology. But some ISPs won't support you if you don't use the provided router (though if you know enough to purchase your own router you might not find such an ISP's tech support much help anyway).
Not really a backdoor, just remote administration.
http://www.lightreading.com/tr-069-still-sexy-after-all-thes...
Now combine that with the typical user's tendency towards password reuse...
"...What?"
I was furious. Time Warner had left a backdoor in all their modems that gives them administrative access to my private connection. And yes -- she did alter the password remotely. She didn't seem to think there was anything wrong with this. I tried googling for relevant information, but wasn't able to find anything more than speculation at the time.