If so, come out and make that accusation explicitly. It would be quite a story, especially if you had a reference to cite.
If you're being a responsible provider, you will be doing backups, in case your primary data dies. But backups are not all that useful if they can just be wiped out by an online process. There's a potential secret copy lying around.
Another possibility is that they run a caching system, that store cache on faster but still persistent storage (or even in memory on systems that don't reboot often). Does their caching setup ensure that it tells all involved caches to delete their copies of the data?
Many systems do some kind of transcoding, generating thumbnails or more highly compressed versions of uploaded video. Does requesting that a file be deleted immediately delete every file it was based on or derived from it?
Furthermore, when you delete data from most filesystems, it doesn't actually get overwritten, just the reference to it is removed. So even if you do delete all the copies that you know about, there's a good chance that the data is still on the disk, ready to be accessed by an off-the-shelf data recovery/forensics tool.
If you're a provider that runs a large amount of storage, you likely run through hard drives at a fairly regular rate, once they start throwing enough SMART errors. Do all providers properly securely erase failing hard drives before sending them off to recycling? Swapping out a failing drive and sending it off to recycling without securely erasing it first likely leads to other copies floating around in the wild that the provider no longer even has access to erase.
I would be more surprised to find providers that actually did securely overwrite every copy they had of a given piece of data when you asked for it to be deleted, than I would by a revelation that providers that "secretly kept a copy around".
Why should you trust them? Because they say so?
And even if they intend to really delete the data instead of just setting the hidden flag - how do you know that one of their employees isn't a "bad seed" and has some weird hobbies like scanning through the Dropcam servers for nudes?
A little paranoia and mistrust isn't a bad thing when it comes to personal data. You can't really control the whole chain so your only option to keep data private that should be kept private is not to hand it out in the first place.