Taking your trusty mountain bike out on the trails is always a good thing to do. You're out in the forest, or perhaps an urban/wildland setting, and it's pedal time. Whether you are on a hardtail or a fully sprung unit doesn't matter. It's just good being out there.
There are exceptions though. The recent thunderstorms that have graced the Sierra have dropped quite a bit of rain in the mountains. It is wonderful to have rain right now, in any amount. More is better, and that looks like what's on the way for the next few days.
What the rain does to the trails in summer is good. The trails get packed down a bit, they aren't nearly as loose, and the dust factor drops significantly. The forest has a great fresh smell, temperatures are cooler, and it's just a great time to ride.
The exceptions? When the rain is significant enough to really douse the trails, things get downright muddy. Mud puddles are all over the place, creeks start to run, or run faster and deeper for a bit, and those knobby earth grabber tires on your bike are put to the test.
Therein lies the problem, and the always present questions: do I ride, are the trails too muddy, do I ride around or through the mud puddles?
The wet ground, if it is sopping wet, can take a pretty severe beating from your knobby tires. Just walking on a really wet trail causes damage. You and your bike are likely to cause even more.
All those ruts that you cut just create channels for erosion. To a large extent, simply being out in the forest causes erosion, even on the best of days. Wet conditions just make it worse.
The mud puddles figure into the rides as well. The best course is to ride through them. Doing so keeps the single track single. Going around, for whatever reason, simply creates a new, and unnecessary track. Think erosion again.
If you don't want to get muddied up going through one of those puddles, slow way down, just enough to stay upright, and go through them.
As for the trails being too wet to ride: if you have to ask that question, the answer is yes, they are. Stay off really wet trails. Too much damage occurs from too many bikes riding really wet and muddy trails.
The trails don't repair themselves. Various mountain bike groups not only build them, legally, they also maintain them. The Tahoe Area Mountain Bike Association regularly holds volunteer trail repair/build/maintenance days. Other groups in other areas do the same.
Riding with concern for the trails is smart. Trails that are abused, in any way, can get shut down. It's better to ride well on good days, while continuing to enjoy the entire mountain bike experience.