Thanks for all of the suggestions. Here's what I found by looking at Nights: Lucid Dreaming, disc one:
1-01 Panzer Nightopia - 192kbps - doesn't work
1-02 Dreams Dreams - 192 kbps - doesn't work
1-03 From Dusk til Dawn - 192kbps - works
1-04 Lies Within Dreams - 192kbps - works
1-05 Reverse Rain - 192 kbps - doesn't work
...
It appears that all of them have a bit rate of 192kbps, which appears to be the same as the tracks in the main remix list. Next, looking at Heroes vs. Villains, I found that each of them had a different bit rate, usually above 192. Since none of these have worked, I figured that 192 might be the cutoff and that higher values wouldn't work. However, I then discovered that track 06, Zeus, track 09, Mega Man, and track 13, Link, all had bit rates lower than 192, specifically 185, 166, and 185 respectively.
I next checked to see if any songs higher than 192kbps successfully played, and I found that a number of the working songs in Pokemon: The Missingno. Tracks had a bit rate higher than 192kbps. Specifically, track 1-11, Drowning Blue, has a bit rate of 224 kbps, track 1-12, Spume, has a bit rate of 226 kbps, and track 1-15, Blue Haze, has a bit rate of 217 kbps, all of which play without a problem.
While it could still be a bit rate issue, these results certainly don't provide an obvious answer. I'll try a few different programs to burn the CDs tonight, and if that doesn't work, I'll try out some different CDs.
Thanks again for all your help!