Tom Desmond wrote (in part):

My house is old, and the
ungrounded wiring, (consisting of just a hot and
an neutral conductor
without a separate grounding conductor), is
deteriorated to the point that
the insulation on the conductors will break and
fall off if not handled very
carefully.

Turn the circuit OFF and call a qualified electrician. Your problem is
not DIY! Old BX cable is known to rust and cause fire hazards because
the rusting causes the impedance of the armour to be so high that the
breakers or fuses won't trip. The symptoms that you describe may be an
indication of such. The problem may be something as simple as a bad
connection, but, IMHO, you need an electrician. I have a master
electrician's license in two states, both of which would condemn your
home if code enforcement caught wind of it. Not trying to be negative,
just safe.