Servos stop after a while in case of 8 servo motor control
If Adafruit say to use the kit to drive multiple servos you need to add a cap and without the cap you can't drive multiple servos, then yes your first port of call should be to add the cap. 

However that's probably not all that's wrong.  If it runs for a bit before dying then it could be a number of causes.  


Not enough power.
Something is overheating.
Code problem.


Use a decent transformer with more than enough amps to power your motors.
Use your nose, does something smell hot?  If you use your fingers you will get burnt!
We can't fix invisible code.