Either of you tax filers may claim any dependent child that the other does not claim. Therefore, what you have described (i.e. for each of you to claim 2 of the 4 dependent children) is legal and dandy. Your dependents can not be claimed more than once, and they must be claimed by someone who actually supports them with money or other value or work.
If your boyfriend does not live at your address, he could also file as single-head-of-household for at least one child.
There is a chance that you & your boyfriend have been together long enough to have a legal common law marriage arrangement and, if so, would likely qualify you two to file married-filing-jointly if you wanted to check into that.