They do, but it's hilariously slow. It's so slow that it's often actually faster to wait for the next morning's Caltrain.
SamTrans runs all-night bus service from SF to Palo Alto, route 397. Once per hour, and the ride takes 2 hours, 15 minutes (!). From there you can switch to the all-night VTA route 22, which goes Palo Alto to San Jose. That's also approximately once per hour, and the trip takes about 1 hour, 15 minutes. With the connection time, you can make the overall trip in about 4 hours.
SamTrans runs all-night bus service from SF to Palo Alto, route 397. Once per hour, and the ride takes 2 hours, 15 minutes (!). From there you can switch to the all-night VTA route 22, which goes Palo Alto to San Jose. That's also approximately once per hour, and the trip takes about 1 hour, 15 minutes. With the connection time, you can make the overall trip in about 4 hours.