For 2016, by daughter 23, and son 25, both worked part-time jobs. They each earned over $4,050 for tax year 2016. I believe this is the threshold amount Dependents can earn in 2016. I believe that because of the amounts they each earned, I can no longer list each of them as Dependents on my 2016 tax return. I believe they are now required to file their individual tax returns in order to get back what they paid in taxes, and because of the amounts they earned. Am I correct in this?
Dependents are always required to file a tax return in their own name for income they earn from working. For a typical child dependent (under 19 or under 24 and a full time student) this does not prevent you from claiming them as a dependent.
But it kind of sounds like they have never filed before? They have been required to file in any year their income was more than $6200 (the amount has changed slightly over time). Not filing means that the IRS creates a substitute tax return in their computer system. which may or may not be accurate. And the IRS usually won't send a "you forgot to file" letter if a refund is owed, because they don't mind keeping the money. (They will send a "you forgot to file" letter if they think tax is owed.) If your kids had income over the limit in past years, they should file.