Another one bites the dust

This whole house-hunting thing is getting a little frustrating. As you know, we've been back and forth with what we thought was "the" house this past weekend. I thought this time we did everything right...we looked at it twice, looked into the age of the roof & appliances, researched the area, etc.
We went in with a somewhat low offer ($11K under their asking price) on Thursday night, wanted them to pay closing costs and wanted possession at closing. They came back on Saturday and had come down 4K from their asking price, said they'd pay closing costs, but didn't want to give up possession until 7 days after closing. We countered yesterday at 9K under their asking price and possession at closing. I told the realtor (Judy), that if the possession was the only thing holding up the sale I was willing to work with it. Say they pay us rent for 7 days or something. She said to stick with the possession at closing and a little lower price and make them pick one that they really want.
This afternoon they came back and the same price they offered Saturday, but now they want possession 4 days after closing and do not want to pay rent or any other fees.
Ummm...no. I am not paying insurance, mortgage, utilities, etc., for another family that I don't know to live in MY house for 4 days, no strings attached. There is way too much liability there! Plus, they're obviously not budging on the price. It's not that we don't want to go up to where they're asking, which actually can't.
I understand their thinking with the price. It's a good neighborhood, they're asking an awesome price for the house, we thought it was great. Obviously, just not great for us.
I know there's other houses out there and all that stuff, but it's a downer. This is the 3rd house now that hasn't worked out. We're not even being totally unrealistic! Oh well, back to the drawing board! On the bright side, I have tomorrow off AND I get to pick up the bear tonight!