  Models based on approximation capabilities have recently been studied in the
context of Optimal Recovery. These models, however, are not compatible with
overparametrization, since model- and data-consistent functions could then be
unbounded. This drawback motivates the introduction of refined approximability
models featuring an added boundedness condition. Thus, two new models are
proposed in this article: one where the boundedness applies to the target
functions (first type) and one where the boundedness applies to the
approximants (second type). For both types of model, optimal maps for the
recovery of linear functionals are first described on an abstract level before
their efficient constructions are addressed. By exploiting techniques from
semidefinite programming, these constructions are explicitly carried out on a
common example involving polynomial subspaces of $\mathcal{C}[-1,1]$.
