It's analogous to saying "there is no voltage [magnitude] below 0V". And that thermal noise guarantees you will always have slightly more than zero, no matter how you measure it (except also at absolute zero temperature).
Different particles, and different aspects of the system, can possess different temperatures. A system need not always be in perfect equilibrium; there needn't even be an exchange mechanism between the different parts of the system. (Thermodynamics assumes there will eventually be some sort of exchange, which in real systems, always manages to happen in the end. Over finite time scales, things can be very different: by pumping a magnetic field, you can chill the nuclear spins of any nonzero-spin atoms. The heat energy contained therein is extraordinarily small, of course, but it's also fascinating that the thermal time constant is in the tens of seconds -- for a quantum system that's room-temperature (or not) and quite low energy (~ueV per flip).
Tim