My guess is if it was across any part of the switch is to prevent arcing of the contacts and prolong the switch life. In my 1949 circline desk lamp it had a .006uf cap across the line and neutral, my guess is noise suppression on the line or again some sort of starting arc suppressor. If the other signs you speak of come on when plugged in, and they're late 30s, you could have done very rare magnetic or thermal aux ballasts. Please post pix if possible of the ballasts.
Welcome to the club bulb buddy btw.