It could if in fact the fan is actually broken.... However, fans on laptops don't always turn on when you turn on power to the laptop itself.
Generally they have sensors that turn them on once the CPU or its heatsink has reached a certain temperature. I know on the two I am presently using, the fans don't come on until sometime after I have powered up the laptops and they have been in use for a while. Even then, the fans only run long enough to bring the CPU temperature down to a predetermined level, and then they shut off again.
In order to see if your fan is broken you could stress test the machine, maximizing CPU utilization thereby causing it to heat up purposely, to see if the CPU fan does in fact turn on or not.