How much room is there after an offer? - 8 months and counting

I received an offer as an insurance claims processor and I think the pay is a little low. I have been trying to stay organised about it. I have a competing interview next week. Did it backfire?

I thanked them and said I was excited but hoped we could talk about the pay. They came back with a bit more.

Keep everything in writing and save the confirmation numbers. It saved me later.