What is the bandwidth of an antenna ? and what happens when a narrow bandwidth is used?
There are many different definitions of bandwidth and engineers are used to switching back
and forth between them and comparing apples and oranges to get the correct answer that they
both are fruit.  For example, if one says that the bandwidth of a lowpass filter is
10 kHz, then usually it means that the output power of a signal at 10 kHz is 
attenuated by a factor of 2 (3 dB attenuation)
compared to the output power at DC.  It is not the case that signals above 10 kHz are
blocked entirely; if that were the intent, then the filter would have been referred
to as an ideal lowpass filter with a cutoff at 10 kHz. For commonly
used low-pass filters, 
the output power decreases at the rate of  n dB per octave 
as the frequency increases beyond the 3 dB point, where n depends on the filter
order: sharper decreases require higher-order (and thus more expensive) filters.

Similarly, if your antenna is usable from 2400 MHz to 2588 MHz, then I would 
hesitate at using it for signaling at a carrier frequency of 2588 MHz since
the upper sideband would be attenuated considerably compared to the lower sideband.
You want to be sure that the entire signal bandwidth fits comfortably within
the specified range of operation.