I've had nuts that are filed to deeply many times. Most players think it's the truss or saddles, or evey that the neck is messed up when this is the case. The thing is, at least in my opinion, the way I've been told countless times to check the nut is just not quite right. The truss adjustment and saddle height have a significant effect on how the nut slotting will interact with overall playability and "setup problems." In my opinion, checking slotting on the nut without "isolating" the nut can lead to mistakes on the nut. I'm talking about when they are filed too deeply because the truss was not adjusted right--with too much relief and/or the bridge saddles were too high. When this happens, you'll never have decent playability again. And mistakes with nut slotting are much harder to fix--it's no longer just a matter of adjustment. It's replacing the nut or at the very least "mickey mousing" it by building it up with CA and bicarb.
The way I check the nut slotting is to fret the string at the third fret. If the string is touching or essentially touching the first fret (behind your finger), that slot is cut too deeply. If there is a bunch of space, playability suffers and you'll always have high, stiff action. I'm sure there is an official measurement that you should have, but I've been doing it for 30 years (and worked in guitar repair for several years), so I just kind of know where it should be for my playing style.
But, if it is too deep, you are going to get annoying "backbuzzing" when you play up the neck (strings buzzing/fretting out between your fretting finger and the nut). The only way to stop it except for over-relieving the neck with the truss (and compromising playability) is to install a new nut or try to build the slot up (a mickey mouse and temporary way of doing it). The upshot is to be VERY careful if you are going to do any filing on the nut. And use nut files; don't try to get away with Home Depot files that are not built for nut slotting. With a high-value instrument, I'd have a repair guy (who knows basses) check/do it.