Obvious situation would be if you don't have enough funds to complete the 'full' double. BUT, if you are an AP (card counter) you should never put yourself in that position. Never start a session or shoe without enough funds to complete it and that includes double downs and splits. We card counters get those juicy plus count, advantageous situations so infrequently, so the last thing we want to do is walk away mid shoe from a strong plus count and advantageous situation.
There are some that are employing a "cover" type strategy of doubling down for less (usually much less), on non advantageous double down situations. One of the common one's is 12 vs 2. Players are doubling down on this for less, usually a very small amount like $5 on a much larger bet (some even go as low as to double for $1). The idea is that this play, which often the dealer must announce, will disqualify them in the eyes of the pit as a card counter.
Doubling 10 vs 10 is another one that is sometimes used for cover. While double down is the correct play at higher true counts, it is incorrect to double at most counts, but is close enough that it doesn't cost that much if you are doubling for significantly less.
I am not a fan of this particular idea of cover. I think unusual plays are just as likely to draw attention as deflect it.