Any known cases of lucky collisions resulting in security breaches? This isn't a deep technical question, but I feel like this is a good place to ask. Modern cryptography generally rely on statistical chances of collisions according to current computing powers. Are there any known cases where someone was lucky and cryptography was broken?

Also, would it be an interesting thing to do to gather as many keys as possible and try to break them all. Would it increase the chances of breaking some of them? I could see organizations like the NSA do something like this.
There have been similar "security breaches" where someone did indeed gather a lot of public RSA keys and successfully factored some of them using Euclid's algorithm.

I believe the weakness was that a lot of implementations didn't use "random enough" RSA key generators (by using a cryptographically secure random number generator) and this resulted in some common factors of said public keys.

Here's the paper with more information if you're interested.