The performance of a communication system depends to a great extent
on E_b, the amount of energy received per bit. This energy
is the integral of the instantaneous (received) power per bit,
or, if you like, the average power over the bit duration times
the bit duration. So, if the bit rate is increases, the bit
duration decreases and so does E_b. Since SNR or SINR is the
ratio of the bit energy to the noise or interference-plus-signal
energy, the SNR and SINR decrease as the bit rate increases.
If you like to think in terms of power and bandwidth, and think
of SNR or SINR as the ratio of signal power to noise power
etc., then consider that decreasing the bit duration means
using shorter pulses, and so more bandwidth is needed. As a consequence,
there is more noise in the receiver since it needs to have
larger bandwidth to capture and process the wider-band signals.
Thus, while signal power remains fixed as the bit duration
decreases, the noise power increases, leading to the same
conclusion as before:
As bit rate increases, SNR and SINR decrease.