Why can't pigtail 14 AWG wire to 20 AMP circuit ?

I was told you can't pigtail 14 AWG conductor to 20 AMP or 30 AMP circuits.What I mean is say you want to get a branch off a 30 AMP outlet and install a new 15 AMP outlet from that 30 or 40 AMP outlet.Why can't I use a 14 AWG wire? Inside main panel, all circuits 15 or 20 or 30 AMP are branched off 100 or 150 AMP ,so why is is o.k. there?? Please respond if you know the answer,don't lecture me from NEC codes ,thanks.