If that's true, I'm in awe of their integrity. Skype on the other hand now has complete disregard for user privacy. Skype stores your voice mails and video messages forever[1]. This is something that they started doing 2-3 years ago and few people seem to be aware of it. It's amazing how low Skype fallen from its early days when it was considered a beacon of privacy and on the cutting edge of encryption and security.
[1] Details: Clicking on Preferences -> Privacy -> Delete history (OS X) or Options -> Privacy Settings -> Clear history (Windows) pretends to delete the voice/video messages but it merely hides them from your view. If you re-install Skype on the same computer or run Skype on a different computer, all those "deleted" voice mails and video messages re-appear. The delete and clear buttons are basically lies; there's no polite way to put it.
Very old voice messages are also accessible even if "deleted".
Text chats do seem to disappear, but at this point I don't believe anything Skype says. I figure they keep the chats forever as well.
Edit: yes, the chat backup functionality still works. I have never tried to restore it though.
Go to settings > chats >> Chat backup to start backing up your messages to Google Drive. (Make sure you have 2FA on your Google account)
It's a "feature" of the normal chats/channels/groups/supergroups of Telegram that you are able to download them to other devices, or to restore them on a freshly wiped device, because (and I'm over-simplifying here, but the end result is the same) they are encrypted with a key known to the server, and which other devices signed into your account can then be authorized to use.
But, the NSA approach to data collection is basically to vacuum it all up and, if possible, decrypt it later. This has two implications:
1. Metadata. "We kill people based on metadata" isn't a joke, it's a quote from Michael Hayden, ex-executive in both the NSA and CIA. End-to-end encryption doesn't hide who you're talking to or when.
2. It seems unlikely that it will be computationally possible for them to decrypt all traffic, but it would only surprise me a little if AES128 is breakable for high-value targets in the next 20 years: increased computing power, better multi-threading, better cryptanalysis algorithms, maybe quantum computing or some completely unexpected technology; it's hard to say what will come along.
In short: mining encrypted data still matters.
I don't see what they can gain from storing masses of old chats and then not allowing users to download them onto new devices. If they kept chats to do analytics on, there's no reason that they wouldn't expose it to users too.
(This may also explain how they survived as a company for so long with so many users and so little revenue. All they're doing is running a few fast servers to shuttle messages back and forth, no storage requirements at all).
On the other hand Telegram does seem to store conversations - if you log in using a desktop app, it will pull down your recent chats.