If someone else claimed your dependent inappropriately, and if they file first, your return will be rejected if e-filed.
You would then need to file a return on paper, claiming the dependent appropriately. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your dependent was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.
If you entitled to claim a dependent you would get the tax benefits; not entitled filer gets to pay the IRS back with penalties and interest.
