As the other answers said, this could also be a capacitive-coupling issue, which is not the pickups' fault but to blame on the guitar's shielding. However, it may just as well indeed be magnetic coupling, which is indeed all about the pickups. From your question, it's not possible to say which one. Here's a couple of distinguishing points:
- Magnetic coupling is much more effective at low frequencies, so if it's a dull, “distant”, sinuoid humming, that's almost certainly magnetic. OTOH, a thin, “bee-like” sawing noise is more likely to come from capacitive coupling.
- Magnetic coupling works predominantly for magnetic fields aligned in direction of the pickup axis. Therefore, rotating the guitar about the neck axis will usually give a periodic change in hum amplitude, at some alignment it should vanish almost completely. Whereas capacitive coupling tends to come from quadrupole- and octopule fields, for which turning the guitar merely changes the timbre of the noise, but you won't find a truely silent position.
- Capacitive coupling can be shielded with any decent conductor (whereas to shield from magnetic fields you need either high-permeability iron alloys, or very good conductors like thick aluminium, which isn't practical). That includes even somewhat poor conductors such as human flesh, and for this reason you can often strongly reduce capacitive coupling by simply grounding your body whilst holding the guitar. In most guitars, this works simply by touching the strings, but you can also try touching a piece of exposed metal on a radiator or water tap. If that makes a significant difference, the hum is capacitive.
- Magnetic interference comes mostly from linear, non-toroidal power transformers. Such transformers are found in most guitar amps, so magnetic hum is often worst when you're close to the amp. Most other modern electronic devices don't have such transformers anymore; other strong sources include electric motors and CRT monitors/TVs, but I reckon that's unlikely in your case.
- Capacitive interference comes mostly from hard-switching semiconductor devices. The worst offenders are triac-dimmed incandescent lights; fluorescent lights are also quite bad. Less strong, but also annoying are many modern switching power supplies, LCD screens and capacitive touchpads. So if the problem is worst next to your computer, it's most likely capacitive (unless you're still using a CRT screen).
Capacitive coupling should be attacked with shielding in the guitar. Cladding the interior with tin foil (must be connected to the jack's ground) works very well in my experience.
Humbuckers are specifically designed to cancel magnetic coupling, so if the PUs are decent humbuckers then the magnetic coupling shouldn't be an issue. If it is nevertheless, the PUs are probably faulty. If you have single coils and suffer from magnetic hum, I recommend you get some SC-width humbuckers or stacked humbuckers.