Last night I had a double-tapped breaker fail. Turns out the former
owner put most of the overhead lighting, and the living room outlets,
on these two taps.

It's a Cutler-Hammer box so the double-tap was legal.

The basement overheads started flickering as I was working, and soon
went out. When I got to the panel, I could smell something was wrong,
so I started feeling breakers for heat. Found a warm one that sizzled
when I touched it.

Luckily I had purchased a tandem breaker (two separate circuits that
fits in a single breaker slot) back when I took possession of the
house.

Is there a limit on how many tandem breakers can be installed in a
breaker panel? There are two now, and at least one other double-tapped
circuit in the house.