The key point is low cost. I am able to get this service for $300 per board but that is not good is it?
Some important bits of information would help us all...like;
1. Where are you located?
2. What is the size of the board, and number/complexity of components?
3. Why are you unable/unwilling to assemble a small batch of boards yourself?
If you are looking to prototype boards for design verification and troubleshooting its not going to be worth any assembly house's time to setup a line to run three boards and you're likely going to pay for complete hand assembly.
If you are doing a test run to verify the assembly house with a large run to immediately follow I doubt they would put up much as much of a fuss. As in "Run 10 boards and is they come out good press the green button and make me the other 9990".
They are making money off volume and you are saving money amortizing setup cost over hundreds, or thousands, of boards.