Generally, MPC protocols have two main parameters: the number of participants $n$, and the amount of corrupt parties you wish to tolerate, $t$. It is typically the case that, as you mention, protocols work by secret-sharing data in a way that at most $t$ participants cannot recover it, but $t+1$ parties can. As a minimum, you will need $t+1$ parties for your computation, since otherwise a set of $t$ corrupted parties could finish the computation on their own and learn the results/secrets.
Some protocols (known as dishonest majority protocols) are designed with $t=n-1$ in mind, which means that you cannot do any better than having all parties at all times. However, some other protocols (in the honest majority setting) consider lower thresholds such as $t<n/2$ or $t<n/3$. This is useful since this can lead to more efficient protocols and stronger notions of indistinguishability (e.g. information-theoretic vs computational security), at the expense of tolerating less corruptions. In these cases you can consider optimizations where, for example, all $n$ parties actively participate in the protocol for some time (for instance, in a preprocessing phase, generating multiplication triples—in case you're familiar with the concept), and then only $t+1$ can remain for the rest of the execution.
