My 30 pint Comfort-Aire Dehumifidier is acting up. When I start it, the fan turns on, the compressor then turns on, and it runs for about a minute, then both the fan and the compressor shut off, and the unit does not turn back on.

Usually in morning the water tank is 3/4 full - this a.m. there was nothing in it (meaning it did not run all night). Is this a sign the compressor is going bad?

There could be several reasons. This site has a lot of info on the Comfort-Aire Dehumidifiers. Did anyone change the humidistat reading to higher? That would cause the unit to not run as it normally would / not collect water. That is obvious - and you likely have checked it. The humidistat itself could have gone faulty as well. It also depends on your moisture level in the space - is it dryer at this point in the season?