I suggest that you try a logarithmic scale for the y axis (e.g. 20*log10(y)).
From your new data (using a hannig window) I get

(scale of y-axis is normalized to 0dB for the frequency with the highest amplitide)
Your harmonics decay faster than for an ideal square wave. At about 15 MHz they drown in the noise floor.
As you noticed yourself, there are signal components in the 210..250 MHz range which stick out from the noise floor, too. I guess they are caused by the ringing/oscillation (1) at the rising/falling edges of your pulses. But since they are close to fs/2, and even increase towards fs/2, I'm also wondering whether the sampling theorem was obeyed, so I'm rather reluctant regarding too many speculations. At least for comparison, you should try to sample them at a higher rate.
There is also a small peak near ~27MHz which I can' associate with anything yet.
The manual stitching of (equal) waveform fragments may have disturbed the results either to some extent (-> add discontinuities, make sampled noise correlated, on the other hand add some noise-free segments (which were noisy in reality), no jitter,...). You should really try to sample a couple of periods from the real signal.
EDIT:
(1) Zoom-in: Ringing/oscillation at edge of your pulse

Looks indeed as if the sampling rate might not suffice to capture the fast oscillation.