If you want to use a faster crystal speed, let's say 20 MHz, you also need to alter the configuration fuses before programming your PIC. The most useful way, at least to me, is to set them in your code.

For the 16F876a, make sure the part number finish with -20, unless it's possible that it doesn't work... even if many here, including me already tried it with nice results.

To use any crystal value >4MHz, you need to set the OSC mode to HS. I suggest you to read the following thread... at VERY LEAST the first 5 posts.
http://www.picbasic.co.uk/forum/showthread.php?t=543

Now, for a 20MHz crystal and for a 16F876a, using MPASM, your code header should looks like
Code:
@        __config _HS_OSC & _WDT_OFF & _BODEN_ON & _PWRTE_ON & _LVP_OFF & _CP_OFF & _DEBUG_OFF

        DEFINE OSC 20
HTH