Police get the physical address from the ISP. The ISP knows which IP was assigned to each customer at any particular time because they log it. Yeah, it's theoretically possible they don't keep the logs, but IRL they always do.
That isn't the source of the uncertainty though. It's part of it in the sense that cost-cutting ISPs may have inaccurate records, but the fundamental issue is that IP addresses are completely unauthenticated.
Some people turn off their modems when they're not using them. One way to steal internet service on some ISPs is to clone the MAC of your neighbor's cable modem (and then the ISP thinks you're them).
People have guests who use their WiFi. Some people use WEP, or weak passwords, or are just nice people who intentionally leave their WiFi open for public use.
Anyone who controls (or compromises) a router can spoof any IP address that routes through it. Same with compromised client devices that can be turned into proxy servers.
Even so, the probability that any given traffic is actually from the person whose name is on the bill is still pretty good, unless that traffic is illegal. Because people engaged in illegal activity have the incentive to do one of those things or any of a hundred other ways to get an IP address not tied to your name.
And it isn't that hard to do those things. Which means there is a high probability that an IP address associated with illegal activity belongs to an innocent party rather than the perpetrator.
See if the illicit traffic to that IP address is really going to that residence. See if the traffic pattern looks like the traffic going in is coming right back out and being relayed somewhere else. Put a van full of government WiFi hardware on the street in front of the building and if the relevant traffic correlates to traffic to a wireless device, triangulate the exact location of that device and see whether it's actually inside the residence. See if anything at that IP address is in communication with C&C servers that imply it has been compromised.
Any of which can be done without shooting a single innocent dog.
Simply making a phone call is not illegal even in the case where that phone could trigger an explosive device.
Remember the phone companies are immune from prosecution in these cases because they're a common carrier, something that also applies to internet service companies now.
I don't know much (or anything) about the prevailing criminal justice philosophies among the judiciary but this sounds like a question that can only be answered by a thorough review of the criminal code and precedent. Wire fraud, for example, is one of those felonies that you can be guilty of in the process of committing other more serious crimes so obviously some components can be broken out of the larger crimes.
That's broadly overestimating the reliability of IP addresses, though. Consumer IP addresses change on a random basis, based on whatever obscure policies an ISP has set up at the time.