We learn from Mesechet Bechorot 30b that you are not allowed to convert someone if they don't accept one detail.
Rabbi learning with his student
But I remember learning from Responsa Achiezer that if you are already converted and then fall back in to your old ways, you are still considered Jewish.
Then we learn from IBid that if someone really wants to convert but may not be able to do one thing we still convert them.
Really? Even if they aren't sure if they will be able to keep everything?
Yes, although we learn an example from Yoreh Deah that someone wanted to convert but a holiday was coming up and new she was going to have to work over the holiday. But, even she still kept all the other rules, her conversion is not serious.
The Chemdot Shlomo says then that it means if someone fails to accept all the mitzvot their conversion is no good. But what if the Beit Din forget to inform them of all the rewards and punishment?
Then that conversion isn't either good. Although, we learn from the Shulchan Aruch that once you convert you are Jewish. Its still a little bit controversial today.
Also, when I was learning about this topic with R' Moshe Feinstien he added in that if a convert doesn't want to give up their life for the "Big Three" they are still considered Jewish.