I'm sure someone has gotten to this before I have but I felt it important information and something I should post regardless.
My friend told me today that after placing his chocobo on the "Rest" care plan, his chocobo didn't wake up at the end of the cycle. It remained permanently asleep. I called a GM about this, and GM Landisl responded to my call. Apparently it is a known and very real game glitch in the chocobo raising process. Landisl didn't have a lot of information to give me aside from that. He wished me luck in raising, and referred to this as a morbid sort of chocobo SIDs... So apparently SE is in the process of finding a solution to this glitch... But until they do it's recommended you do not place your chick on the "rest" care plan. You may want to hold off on raising for awhile, or use the "basic care" plan to give your chocobo some rest without actually resting it. The glitch is not a 100% of the time thing... but I personally don't want to take the chance.
To those who've already placed their chocobo's on a rest plan, I pray for the safety of your chick. My fiancee put hers to rest tonight and wasn't able to get back to San d'Oria in time to change it. She went to bed very upset with this information.