Keeping things simple, if the input data SNR is, say, 16 bits, and the PID coefficients are 16 bits, then there is 32 bits of significand at the output of the control gain multipliers. For single precision float, this is truncated to 23 bits. So, the system may settle when the transducer input is near zero (and the exponent can scale down to maintain sufficient precision), but then fail to settle well at the ends of the transducer range (where the transducer input is near +/-2^15 and the exponent in the single needs to stay high to avoid saturation). Also, consider the numerical integrator is running an infinite series of adds, so therein lies a trap for error growth, as well.

I have made this mistake before, and the sometimes strange control behavior can be frustrating. Considering the small performance penalty for using doubles on modern hardware, it is excellent advice.