Can they - yes- there is nothing to stop them.
The IRS will process both returns and pay any refunds. Shortly (within a year) the IRS will mail letters to both taxpayers asking if their tax return was filed in error and suggesting that they amend if they improperly claimed the child.
If neither taxpayer amends, the IRS will send a second letter asking for each taxpayers proof that they are entitled to claim the dependent, such as proof that the child physically lived with them more than half the year. School records, child care records, household receipts, medical bills, etc., that show that the child lives with you should be retained.
The IRS will evaluate each taxpayers claim and award the dependent to one taxpayer, the other will have to payback any refund received plus interest and possible penalties. The taxpayer that had physical custody usually always wins.
Do not ignore the letters or you will loose.