Doesn't sound like you're unit testing well then. I'd take good unit tests over integration tests any day. Integration tests are inherently flaky.
If I had a dollar for every time I asked someone why they weren't fixing their shit, and they answered "Oh, I thought the build was failing randomly again." I'd have a lot of dollars.
Yes, there are bugs that look like bad luck, but are really flaws in your logic. I have fixed a lot of bugs that look Random, but eventually you're left with, for instance, selenium just refusing to click a button.