You can definitely have a tube amp with no audible hum. Requires both sufficient filtering of the B+, as well as any fixed bias references if present and filaments if DHT. Also, you have to worry about EMI on the OPTs. My tube amp has a bit of audible hum with headphones more than about 98dB/mW sensitivity, which goes away if you move the OPTs an inch or so further away from the PS transformer and choke, at least listening with 109dB/mW sensitive Shure E4. There are certainly noisy tubes due to construction faults or damage, h-k leakage, but with good tubes and a well designed amp, it is definitely possible to have a tube amp that is dead silent, even with IEMs like dinny was talking about. Its just a real pita.
Edit: what Nate said.