It is really not a big deal nowadays, the problem of the same scale as having index out of bounds error (no language has good defence against this, yet it is not a catastrophe).
Unfortunately, no, they don't. Not after your application has been running for a while; newer JVMs arbitrarily decide you don't need the stack trace anymore, and all you see in your logs is "NullPointerException" (unless you still have the logs from several weeks ago, just after the last JVM restart, which might still have the full stack trace). Older JVMs were better, since they always had the full stack trace; debugging NPEs was easier with them.