Then you're either a)not testing correctly or b)writing code with too many dependencies, or c)both. If a thousand tests break from a single code change, this should be a code smell that says your architecture needs work.
This is where testing gets its real value. Finding bugs is just a side effect of testing. The real value is that testing helps you design good code.
I think he means the reverse though. It's not that a single line of code causes thousands of failures, it's that you'll never get that particular test to fail unless pretty much everything else is failing to, ergo it's a useless test.
I don't necessarily disagree with your point, just clarifying.
This is where testing gets its real value. Finding bugs is just a side effect of testing. The real value is that testing helps you design good code.