Only to a point.
The 5-6m of RG-174 cable included with most portable active antennas is actually a required part of the "antenna system". The 6 dB or so of loss contributed by the cable is needed to lower the signal power into a range usable by the receiver.
For instance, typical active GPS antennas have an LNA gain of 28-30 dB while some receivers have a MAXIMUM external gain specification of 26 dB to 33 dB. Simply cutting off all but a foot or so of the cable results in the R.F. front end of the receiver being overdriven.
The receiver will most likely still operate, but Time To First Fix (TTFF) and tracking performance will be degraded. The recommended minimum length of RG-174 for these types of antennas is 3m, and the maximum is 12.5m. If you are designing a project where the antenna is going to be in close proximity to the receiver and a lot of extra cable is undesirable, you should consider using a receiver fitted with the onboard LNA coupled to a passive antenna.