Look at Jimi Hendrix. He had to tune a whole bunch because of his obsessive use of the tremolo.
The tremolos on old Fenders really aren't the best.
There have been many improvements to tremolo bridges since Hendrix's day. It's uncommon to see tremolos pivoting on six screws. Almost all modern tremolo bridges pivot on two studs. This helps the bridge to pivot smoothly without binding up. Also, the bent-steel style saddles on old Fenders provide a flat surface for the strings to rest on, which allow them to shift side to side. Most modern bridges, tremolo or hardtail, have a little groove for the strings to rest in, so they cannot shift around.
Furthermore, heavy tremolo users often opt for locking tuners, or if you have a bridge like a Floyd Rose, with fine tuners on the saddles, a locking nut. I don't believe I've ever seen locking tuners or nuts on a bass, though.
Aside from stretching the strings when you pull the tremolo arm up, if your bridge allows for it, the key to staying in tune is to make sure that the strings can return to pitch properly by eliminating friction. Be sure that your nut is lubricated properly, so that the strings cannot bind in the slots.
Also, keep the number of wraps around the tuning peg to a minimum. (On guitars, you tie the string in a sort of knot around the peg, but bass tuning pegs are very different than guitar tuning pegs, so you can't do that.)