Your Chocobo is Spoiled!
Your VCS trainer will notify you that your Chocobo is Spoiled Rotten.  The only thing you can do is Scold your Chocobo :(.  It also make take 2 Sessions of Scolding before your
Choco snaps out of it.
Since Scolding lowers affection, i suggest focusing more on Greens than Carrots during this situation.
During this time, your Chocobo's Stat increasement has declined.
Remedy:  You must Scold it.

Your Chocobo wont wake up!
This situation really doesn't happen all that much, and mostly happens after you've set your Chocobo on a Resting Plan.
 
The Resting Plan, in my opinion, should only be a last result used when your Chocobo has run very low on both HP and affection.
I have never had to resort to this, as there are plenty of other ways to handle low HP and affection.   Cutting Carrots out of a Chocobo's diet and feeding it just greens and taking it on Long Walks should drastically improve the HP and affection of your Chocobo after a few days.  Even more affective is feeding it a Celerity or Tornado Salad for a few days, taking it on 1 Long Walk for Affection Purposes each day while leaving the rest of its HP untouched.
 
In any case, if your Chocobo wont wake up,
Remedy:  Scold it  and/or
                give it a Chocotonic

Your Chocobo is depressed.
This results in quite a while of not looking after  your Chocobo.  If this continues for too long, and I do stress Long since this condition usually only happens when you stop looking after your Chocobo, your Chocobo will Run Away.  Subsequent Run-Aways results in Auto-Retirement.
After, you will be able to call your Chocobo with your Whistle.
Remedy:  Start looking after your Chocobo.
                Feed it Greens or a Celerity Salad.
                Take it on Long Walks many times.



Advice for Keeping your Chocobo Healthy:
1)    Keep your Chocobos Affection up at
Regards as Parent as much as possible.
This is a very useful piece of advice anyways because the higher your Chocobo's Affection, the more responsive it is to Stat Advancement.
The best ways to raise Affection are by
Feeding your Chocobo Sharug or Azouph Greens, and taking it on Long Walks.

2)    Try not to overwork your Chocobo.
This,in turn causes injury as well as lowers Chocobo Affection.  If your Chocobo is on a Rigorous Careplan, make sure you feed at least 2 Greens a day and try and take it on 1 Long Walk a day as well.

3)   Implement Medicinal Greens into your Chocobos Diet, especially if you notice your Chocobo's health declining frequently.
This should prevent your Chocobo from developing a Malady.  For Chocobo's undergoing a Mental Agenda, a Tokopekko Wildgrass thrown into the diet now and then keeps your Chocobo healthy as well as slightly boosts Discernment.

4)   When your Chocobo becomes sick in general, feed it the corresponding 2 Medicinal Greens instead of 1.  This should get rid of the sickness and prevent it from lingering on.

Your Chocobo can Catch a Cold [ill]
Your VCS trainer will notify you when this happens and will also notify you that you must feed your Chocobo Tokopekko Wildgrass in
order to recover.
During this time, your Chocobo's Stat increasement has declined.
Remedy: 1-2 Tokopekko Wildgrasses
               an Herb Paste
               a Celerity Salad
Affects:    Medicinal Properties
               Slight Discernment Boost
Note:       A developing high fever will result if
               ignored

Your Chocobo can Injure Itself :(
Your VCS trainer will notify you that your Chocobo has become injured and your Chocobo will appear injured in the Stables as well..
Your VCS trainer will remind you to feed it Gausebit Grass.
This is frequently due to overworking it in Careplans.  The most common Careplan that injurs a Chocobo is Carrying Packages and Delivering Messages.
During this time, your Chocobo's Stat increasement has declined.
Remedy:  1-2 Gausebit Grass
Affects:    Medicinal Properties
Note:       May take 1-2 day-sessions of
                feeding before it takes affect.

Your Chocobo has a Stomach Ache [Sick]
Your VCS trainer will notify you that your Chocobo has become sick and will advise you
to feed it Garidav Wildgrass.
During this time, your Chocobo's Stat increasement has declined.
Remedy: 1-2 Garidav Wildgrass
Affects:    Medicinal Properties

Your Chocobo has become Lovestruck
Your VCS trainer will notify you that your Chocobo appears to be in love. In the stables, it appears to be physically sick with hearts floating over its head.  It's important to try and fix this state as soon as possible because it can end up lasting for multiple days, perhaps even up to 4 days.
It is speculated that Resting, Scolding, and Taking your Chocobo on Long Walks can cure Love Sickness on Rare Occasions.
Remedy:  Celerity Salad
Affects:    General Stat and Affection Boost
                and Medicinal Properties

Your Chocobo is Bored!
Your VCS trainer will notify you that your Chocobo is Bored.  They will also tell you that you should Compete it in Races, and that's, indeed, what you should do!
I usually compete it twice in Races, to prevent the "boredness" from lingering on, and pump
my Chocobo up with just Greens for that day
to keep its affection high.
During this time, your Chocobo's Stat increasement has declined.
Remedy:  Compete in a Race