I think this might be a very common problem with syringe users, so thought I would post about how I fixed the problem.
I have only one 1mL syringe that came with a needle and have had it for about two weeks. A couple of days ago, the black bulb became stuck at the end of the syringe. I pushed it out using a large paper clip, but it kept getting stuck when I reinserted the plunger.
So, I pushed a thin wire brush (from the kitchen drawer) through the syringe a few times to remove anything built up inside and rinsed it out real well (also soaked both parts in soapy water the other day).
Then I added a thin layer of vaseline onto the black bulb, and now it slides into the syringe like it should. I've also realized that, when I'm not using the syringe, I should leave the plunger out of the syringe, so it won't keep getting stuck.