Coding counters the uncertainty principle by allowing multiple measurements, which can then be averaged. Thant counters the contribution of random noise.

There are practical signals we use every day that are "below the noise floor" before we decode them.

So while there is an ultimate limit of the maximum coding rate for a given signal-to-noise ratio, this is expressed in terms of a data rate (i.e. bits per second). If you're fine with lowering your data rate, there is no fundamental theoretical limit, as far as I understand.