The circuit breaker is on, but there is no current anywhere on the line.
One wing of the house was made into an apartment in the '40's, and this
was also on the bad line. We had that rewired to a separate panel about
10 years ago - this on/off problem only happened about 2 years ago. So
load shouldn't be a problem since we have considerably LESS drain on the
circuit now than in the past.


Then measure the voltage at the breaker itself (output screw to neutral
bus). if the breaker is on and there is no voltage, the breaker is bad if
there is voltage, you have an open in the first J box it goes to (unless
someone put a splice in a concieled location, now we would know why not to
do that).

Check that the wire nuts in the first J box on that branch are all in good
tight condition. May need to open them (splice at a nut) to inspect wires
for damage or corrosion. If there is no voltage from hot to neutral or
ground at the first J box then you have the unlikely failure of a wire along
its length (possible Nail intrusion)

If you cannot deduce the location of the cable or first J box on the branch,
you will need to use a signal tracer to follow it inside the wall from the
source at the breaker panel.

Here is one reasonably priced tracer that can be used for LAN, phone and AC
(with hot disconnected at the panel). Others are available optimized for AC
if you look around
http://www.stayonline.com/detail.aspx?ID=5241