1) I remember building a game in 90s where the locations of resources on a 2d game map were generated by successive calls to random() (and then were probably mod 1024ed or something). But, when we looked at the locations, they were clearly in a line pattern. Huh? After a while of confused debugging, we eventually discovered that the RNG was a linear congruential generator, which, when used the way we did to generate N-tuples, tends to produce points that tend to lie on a small number of N-1 dimensional surfaces! Oops.
2) I remember a big debate with a statistician at a company we were selling data analytics software to. We were putting random data elements into random groups to support certain types of statistical analysis. He was adamant that the randomization we were doing would introduce bias into the data and showed us similar tests he had done that "proved it". He told us that his company would be dropping our product because they couldn't trust it. I told him that there was no way bias was introduced and I took a closer look at his tests. It turns out the RNG in whatever tool he was using was just not very good and it had apparently shaken his very understanding of statistics :) Luckily, I told him that we were using MD5 for randomness (once bitten!) and that if he simply redid his experiments with MD5 he would see the bias disappear. To say he was skeptical would be an understatement, but, thankfully for our contract, his tests didn't actually crack MD5, all bias was gone, and we got a very generous email from the skeptic informing us that he would be trusting our software for everything from now on.