And assuming you’re using regular displacement rather than vector displacement, only one channel is being used - it can only have 256 elevations in there. That said, sometimes the culprit isn’t the bit depth itself, but how that bit depth is utilized. If the texture shows up from black to white, then it’s as godd as it’s gonna get for that bit depth. If the texture is only from dark grey to slightly brighter, you’re only utilizing a fraction of what is available.
Technically, you could use a 3-channel 8-bit texture to store more information (rather than 1-channel) to get more information in, but it would likely look weird and not intuitive to work with. Alternatively, you could have 3 different height maps in there for different amplitudes and frequencies of detail - together they might be able to camouflage the output from the one with coarsest information. In short - 1r + 0.25g + 0.1*b or something. Even this simple addition gets you 768 levels rather than 256.