Typically, the ground truth is taken to be the continuous model.
I would dispute that point. In e.g. the Ising model (and many other lattice models), the fundamental degrees of freedom of the theory are discrete, and it is the continuum formulation which is the approximation - typically arrived at via some coarse graining procedure. In solid state physics, all physical samples have a finite volume and therefore the electronic states have a discrete energy spectrum, but in the interest of simplicity one takes the thermodynamic limit in which the volume goes to infinity and e.g. the density of states becomes continuous.
The kinds of "errors" that can arise from this procedure depend on how coarse the approximation actually is. One interesting example is when one considers magnetic skyrmions, which are topological solitions in the magnetization field of a magnetic material. The magnetization $\mathbf m(\mathbf x)$ is treated as a continuous function, from which one can calculate the energy of a magnetic configuration. Since there is a term in the energy functional which depends on the spatial gradient of $\mathbf m$, a discontinuity would require infinite energy - which suggests that topological solitons like magnetic skyrmions should be completely stable.
Of course, this analysis discounts the fact that the magnetization is not continuous when you zoom in close enough, and indeed one of the annihilation modes for magnetic skyrmions is that they may shrink to the point where the continuum approximation fails and then simply pop out of existence with the flipping of a handful of spins.
On a more philosophically interesting note, if you study the Ising model on a finite domain, then you will find that there is no such thing as ferromagnetism or permanent magnetization. It is only in the limit as the number of sites goes to infinity - in which the average magnetization becomes a continuously-varying quantity rather than a discrete one - that a magnetic phase transition can occur. This suggests that the existence of permanent magnetism is itself an "error" which we obtain by taking a limit in which a certain quantity becomes continuous.
Interestingly enough, we want to make this kind of error. In reality, there are no such thing as permanent magnets - but if the chunk of iron in my hand will retain its stable magnetization for a hundred trillion lifetimes of the universe, then I would far prefer to make the "error" that the magnetism is permanent rather than to make the technically correct statement that in the limit of infinite time, all magnetization is transient.
