Stp, etc,: a tax question

Hi,stp or other experts,
I have another tax question. It may be naive.
If I and my wife file seperate tax return (using 1040A form), can we both claim our kids as dependents (that means they will be claimed twice), or only one of us (me and my wife) can claim them as dependents?
If both of us can claim the children as dependents, we can get more child tax credit and additional child tax credit. It seems good for low-income family, but is too good to be true.
Look forward to your reply.
Thank you.