FFT of SIN waves with different phase delays
You aren't looking at the frequency content of your sum of sinewave.  You are looking at the frequency content of a rectangular window on your sinewaves, and the window (FFT length) is shorter than the least common multiple of all the sinewave periods.

Signals that are not orthogonal within a window can cancel each other out partially or completely within that window.  What has happened is that you've chosen a window of a length such that each sinewave can get nearly completely cancelled out by the next higher and lower frequency pair of sinusoids of the "right" phase.  The ones on the ends are the only ones that aren't sandwiched, and thus aren't cancelled.