How many is too many mods?

Well, the limit's 256, though that doesn't count simple texture or mesh replacers, just mods that use esm files - including the game itself plus dlc's.

My current game has 189 and is (touches wood) perfectly stable.

depends how many your system can handle and what kind of mods you want to run

not all PCs are created equal

"Too many" is exactly one mod more than is needed to create the game that you want to play.

Like it was stated and the answer will vary based on the mods you are using, what your PC can handle and how stable the game is running for you or I should say what levels of stability you can live with.

This, or one mod more than your PC can handle.