If you are still running the original points distributor, I would check the
capacitor.  Make sure it has good contact with the strap and plate.  It's a
very inexpensive part, replace it, and you can check it off the possible
cause list.

Brian

Check battery terminals and ground strap from battery to frame / motor
to frame.

C3s have some fusible links that can break with age as a fusible link is a
section of wire smaller gauge than the rest of the circuit.  It is possible
that this has happened and as you have road vibration and heat, the wire
"opens" and you lose power.  Then after 30 minutes of cooling, it closes
again and you run.

In some C3s, these are near the horn relay. In others, they are near the
starter.  Check with a voltmeter or test light while flexing the wire.  You
can buy the links at Chevrolet with the kit to install the big black lumpy
splice, or simply install a 3 inch piece of wire one or two gauges smaller
(numerically bigger) than the regular wire.  For example, if the wire is 12
gauge, install a 14 or 16 gauge section.