Can an aircraft cabin have a lower pressure than outside
Yes, absolutely is CAN. It's not desirable, but if the seals are all tight enough & you descend quickly enough, it can happen. Because the effects on the structure would be bad, aircraft typically have a negative pressure relief valve to prevent this from happening. By definition, you don't need to prevent something that can't happen, so yes, a negative pressure differential is possible.
What you want to avoid, and with modern pressurization controllers, you probably can't reach, is a situation where you descend quickly enough to "catch the cabin" and the cabin altitude matches the aircraft altitude. (Which is to say, you're depressurized -- with or without the negative pressure relief valve opening -- a few thousand feet above the ground.) At that point (assuming you're still descending quickly), the cabin rate goes from something mild, a couple hundred feet-per-minute, to something much greater, perhaps a couple thousand feet-per-minute. If you aren't used to dealing with that sort of a rate of change, it is disconcerting & uncomfortable, and for somebody who is flying with a minor head cold who could keep up with a gentle rate of change, it could be pretty painful. And since nobody wants to scare Grandma, this is undesirable!
Thankfully, with modern pressurization controllers, you probably won't ever get there -- they'll manage the rate to keep things on schedule regardless. With an older analog controller, it's possible, although probably pretty rare.
As explained in other answers, the modern controllers start to pressurize so that the cabin altitude goes slightly below the airport elevation (i.e. the cabin pressure rises slightly) at some point before liftoff (when throttles are first advanced, typically -- it's all automatic). And on landing the cabin is likewise slightly pressurized, and that pressure bleeds off slowly after you have weight-on-wheels. This is done so that the system transitions into and out of pressurization gradually, instead of immediately upon liftoff or on landing. The outflow valve moving at full speed can give you a pretty good "pop," and that's not comfortable, so gradual transitions are designed in.
It's all about keeping the paying passengers comfortable & happy!