I just tried part 13-15 again this morning, and it finally worked! It started at a blazing speed of about 100 MB/sec, and ended up averageing about 25, but it got here without timing out.
So then I decided to gamble on the Big One: 21-24, which should be about 457 MB. It started ok but timed out after 12KB, and then I couldn't even connect to the store to login to try again.
30 possible download tries looked enormous before I started. Now it doesn't look like too many at all.