Camping options are pretty good if you're willing to drive an hour or so out of San Jose for your tent camping escape.
Henry Coe State Park is southeast of San Jose - it looks close on the map, but the entrance drive can take forever!
Mount Madonna County Park is one of the best places to camp, especially if you want to make a day trip to Monterey, Carmel, or Pinnacles National Monument. Definitely stay here if you want to camp for the Gilroy Garlic Festival, as Mount Madonna gets sea breezes and is much cooler than the hot valley floor. Expect clean bathrooms, nice showers, and big shaded camp sites.
Sanborn County Park is nice if you don't mind hiking in to your campsite, this tends to weed out some of the trouble makers that urban campgrounds attract.
Grant County Park - east of San Jose - is underutilized, and can be one of the best options for finding a last minute camping spot in the Bay Area. It's got some great hiking trails, and it's not too far from San Jose.
Towards the coast, several state parks have tent camping options near Santa Cruz - Big Basin State Park,