I started having this problem after upgrading from version 1.5.? to 1.6.2. I was able to solve it by Settings -> Configure Kile -> Tools and adding %S to the end of the command lines there. Without the %S, the tools, such as latex are being called without being passed the name of the file to process (%S expands to the base name of the file). Because latex has nothing to work on, it doesn't produce a log file, which Kile is expecting. Kile then complains.
So, e.g., change
Latex latex %S