Side Lobes in the Magnitude Response of a Low Pass FIR Filter I have the impulse response (from the filter coefficients) of an FIR filter obtained from MATLAB's "interp" function using the command:
[y, b] = interp(x, 5); % where b contains the interpolation filter coefficients. 
When I plot the magnitude response of this filter using "freqz" in MATLAB, I get the following

What do those side lobes between 0.5 and 0.7 (x pi rad/sample) and 0.9 and 1 (x pi rad/sample) mean for my FIR filter? Why are they there?
The filter is optimized to suppress signal components in the frequency ranges where the side lobes are low. The high side lobes in are just a consequence of optimizing the stopband suppression in those other frequency bands, where aliasing components are expected. The regions with high side lobes are basically "don't care" regions. If you wanted to increase stopband attenuation there, the stopband attenuation in the other - more important - regions would be decreased.