Solved! Go to Solution.
You'll probably have to contact Spotify support directly for help, but I've been working with them for over a week and the problem continues. I've found some suggestions from other users with the same problem and I've tried them, and some of them work, but I can't make any promises.
1. Try playing one of the songs in the playlist that you're downloading.
2. Close Spotify while it's trying to sync, then re-open it.
3. In your iOS settings, turn off your WiFi, then turn it back on again.
This problem, as well as slow sync speeds, and songs not playing at all, are reported by quite a few users. Nobody seems to know what the cause is or how to fix it.