Do Connected: [blank] cards stop dropping?

Sure feels like it – I haven’t had -any- for several days for the groups I actually want to be connected with. This is making it hard to change a job, for one…

Nope. The RNG tracks your specific needs and wants so it can accurately determine which cards not to give you. It’s just being mean; they’ll show up eventually, as there’s no way to lock them.

If you’re really desperate, you can use Favorable Circumstances at Write Letters in your lodgings to summon the card you’re after. Or you can try and even the odds a bit by culling cards from your deck; reduce conflicting connections, sell pets and items with cards and move to an area with no location specific cards (Like the Labyrinth of Tigers or the Forgotten Quarter) to reduce your card pool and increase the chances of getting the card you want.
edited by Spacemarine9 on 6/7/2014

Still, it’s mighty suspicious that I didn’t get any Connected: cards for the few factions I have at 15+ in a while now…

It’s just bad luck. Hopefully you draw the card you need soon!