Alan,

The suggestion about the bad connection is a good one, but the bad
connection could be in the battery itself.  Here are some things to think
about:

1) The charging system should provide about 14 volts with everything
   in the car turned on with the engine at about 2000 rpm.

2) This charging system voltage should be within a few tenths of a
   volt at all points starting from the alternator output wire and
   ending at the battery posts themselves (not just the clamps).

3) All of the above may check out, but a poor internal connection
   in the battery will show up in the load testing that Tom suggested.

It's only about 68 in Phoenix today.

Mike    '84GL  '60 Vespa  '58 Isetta      Phoenix, AZ