Yes. The bridge is meant to be grounded. There were 2 ways of doing this. One is just a piece of wire. The other is a capacitor and resistor in parallel. They both do the same thing but in different ways. But both will behave differently in a simple resistance test: that's why you really need a techie to do the test.
If your bass still behaves like that at your teacher's house then it's almost certainly the instrument that's at fault.
Is it a Fender / Squier? If so go to
www.MrGearhead.net. That site has all the wiring plans for Fenders (somewhere in the support section I think). It's not uncommon for a control pot to get loose and constant twisting eventually breaks a wire. I wonder whether that might be your problem?
Yes. You may damage if you try the idea. It'll probably be OK but isn't wise to experiment with earth faults.
Believe me, I've had 30 years in electronics / electrics / elec safety.
Get it checked. If it's the bass a music shop should have someone capable of handling the repair.
John