One of two separate lights on a double switch has become “dominant” – bad switch?

Report of electrical/lighting anomaly at grandma's house. Obviously, it will be a while before I can check things in person. Would appreciate diagnosis of most likely problem. (A relevant search-result was not found using the particular terms tried.)

From memory (and forgive any misapplied terminology): 2-gang duplex wall plate (i.e. four cutouts). Duplex outlet on left. Double toggle switch on right. One switch controls a "chrome" circline fluorescent ceiling fixture (12 inch + 8 inch bulbs). The other a simple 24" T12 fluorescent over-sink vanity light. Outlets working fine.

Reported that one switch has become "dominant". If ceiling light is on by itself, toggling the vanity on closes the ceiling light off (vanity light comes on). If vanity light is on by itself, flipping the ceiling-light switch does nothing. Each fixture by itself is toggled on/off by its respective switch when used alone.

Is it almost certainly a bad switch, or could a fixture (bulb or ballast) or wire somehow have gone bad to produce this effect?

If it's relevant: all wiring would be 60-85 years old, and insulation could even be cloth-of-any-color. (And it probably wouldn't be impossible that a light switch somewhere in this house may have been wired to break neutral rather than hot, or not have a ground wire connected.)

Thank you.