My 21 month old sometimes refuses to nap. She will play in her crib but won't sleep. She is then miserable come 4pm but by then it is too late for nap. When she doesn't nap she falls right asleep at night, when she does nap she plays in her crib for about half and hour before falling asleep. I can't figure out if I should just give up on nap and let her get used to it and then maybe she won't be so unhappy in the late afternoon or continue to have her nap. I am worried she won't be getting enough sleep if she gives up her nap. So, when is too early to give up a nap?
You don't have to end the naps abruptly. You could transition away from "nap time" by replacing it with some "quiet time" instead:
The "rules" for quiet time would be simple, something like she has to stay quiet, no talking/crying/singing and no noisy toys; she has to stay in her bed (or at least in her room); she should expect to remain alone; she does NOT have to sleep but she can read, or relax, or do other slow and quiet activities alone. When the daily quiet time ends, she can resume whatever non-quiet activities she likes.
(The idea of "quiet time" isn't mine. I think there's an older question in here somewhere where this was introduced as an answer. I'd appreciate if someone can find it and link to it, to give credit where due.)
In all things sleep for our children, my bible has been Kim West's The Sleep Lady's Good Night, Sleep Tight: Gentle Proven Solutions to Help Your Child Sleep Well and Wake Up Happy (Amazon page). It's the only parenting book that I don't lend out, because I never know when I'll need it on short notice.
Based on that book and my experience, I think 21 months old is too young to give up naps. (Not only because a 21-month-old awake all day will drive any parent nuts, but also for the child's sake!) Our 4-year-old still naps once a week or so, and can nap for 3 hours if we let her sleep that long. Our 19-month-old always naps 2-3 hours, and is quite cranky if she doesn't nap.
Here are the most important things I found for making naps happen (many of which I learned from the Kim West's book):
A very dark room. We had to get blackout shades, and they helped.
A very consistent sleeping routine. I'm almost a fanatic about it. With our daugher it goes like so. We eat lunch, change a diaper if needed (all such awakening activities happen early in the routine!), read a story or two, close the curtains, talk about how it's sleeping time and how teddy (her lovey) is also tired, sing her a song, lay her in the crib under a warm blanket (with teddy), and say sleep tight, and then go out of the room.
A consistent lovey (her favorite teddy bear) that always goes to sleep with her, and participates in the sleeping routine. We don't take this teddy anywhere outside the house for fear of losing it.
The "Sleep Lady shuffle" described in Kim West's book (fading away, I think it's also called). That helped our daughter learn to sleep on her own at night, which helped her at naptime too.
There's lots more discussion about naps in Kim West's book. For really hard cases, I can also highly recommend Kim West's telephone coaching sessions http://www.sleeplady.com/personal-coaching/ . Sabine (our 19-month-old) had been getting up at 5.45am every day, no matter what we did or tried and despite sleeping well at naptime and at night. A phone session was extremely helpful, and now Sabine gets up at 6.30 or 6.45, which is a vast improvement.
(I have no financial interest anything I've recommended above. Though sometimes I wish I did!)
EDIT: With the shuffle, you teach your child to hold you in his/her mind, as you are progressively farther away, and thereby go to sleep calmly. Each stage lasts three nights. In all of them, you put your child in his/her crib when drowsy. In the first stage, you stay right next to the crib making soothing sounds ("sssshhhh", "sleeping time", "teddy is also tired") until he/she falls asleep. Three days later you are in the second stage, and you stay halfway across the room but very visible. The next stage you are across the room. (You can vary the positions, and I am probably not quoting it exactly, but rather am describing what I did.) Then by the door but inside the room and visible. Then somewhere by the door and not visible but audible. Then just outside the room and audible. Then outside the room not too far away. And you're done!
Well, almost. In the last stages, when you are not visible, you come back every 5 minutes if needed to sooth your child if needed (so it is definitely not a "cry it out" method).
We did almost exactly this for the night time sleeping. For naptime we found that we had to skip a few stages, and just go more quickly toward the "out of the room" stage and allow about 10 minutes of crying. Otherwise our presence in the room was just an invitation to keep playing and to avoid falling asleep. Whereas at night, maybe the total darkness made it obvious that it was sleeping time (even with blackout shades, at naptime there was still some light).
The idea behind it is that your child develops more and more confidence in your presence even when you are not there (but it is too hard to do all at once, so it is progressive). Thus, he/she feels comfortable enough to fall asleep.
Unless skipping the nap makes her cranky all afternoon, I'd probably skip the nap in your case. But for our daughter we will still rock her, sing to her or simply take a short walk in the stroller to make her fall asleep for the mid-day nap. Like yours she won't fall asleep by herself.