2000 Chevy Impala engine not starting

After the car sets it does not start.
You can turn the key and nothing happens.
You have headlights dash lights and they do not dim down when you turn the key. Like the you have the vehical in gear and it wont start.
I change the battery, checked all connection on the battery, had the starter rebuilt with a new selinoid
When you try to start it and nothing happens, if you let it set for 5 or 10 minuntes it starts fine and will do so many times. I thought I had the problem solved when I replaced the seliode, it worked great for a week now it back to acting up again and its not every time you go to start it.