I just wanted to say that DD has/is a terrible sleeper.
She is now 11 months and naps (fingers crossed) are a lot better. She has gone from 20-30 mins to 1.5-2 hr naps.
Like others have said, she was clearly overtired. We worked a lot of her being able to take herself off to sleep. It almost verged on CC, but only because she got to the stage where she got more upset when we tried to settle her.
Now we put her down, if she's initially crying, she normally stops after a couple of mins. If she's initially quiet but then starts crying, we go in, pick her up and hold her for a few mins (normally in a different room) and then try again. If she doesn't go straight off, she has shorter naps. This happens if we've put her down too late.
Now if she's up for 2 hrs, she is put to bed even at 11 months. Now if she sleeps less than 1 hr, it is obvious she needs more sleep.
DD even if she's tired, if she's interested in what's going on, she could continue forever without other people being aware that she's tired. Even now DH misses the signs and he's a SAHD so it can be really subtle things you're looking for. For us, crying when removed from something and rubbing not quite her eyes, but her nose but signs that she's tired.
Hope this is of some use to someone.
As for night time sleep, although she goes off easy (never a problem with this) we still have multiple (but thankfully normally short) wakings.