Let $H$ be a set with a binary operation $\cdot _H$ on it. To show that it is a group, one has to show that $\cdot _H$ is associative, find an identity element in $H$, and so forth; it might take some work. However, if we knew in advance that $H$ is a sub-*set* of some group $G$, with $\cdot _H$ induced from the multiplication of $G$, then it would be much easier to check that $H$ is indeed a group.

In a similar manner, suppose that $\mathcal N$ is a subcategory of a model category $\mathcal M$, and consider $\mathcal N$ with the structure induced from $\mathcal M$. Is there any condition that allows one to check easily if $\mathcal N$ with the induced structure is indeed a model category?

EDIT:

As a motivating example, consider the following. Let $\kappa$ be some infinite cardinal. Denote by $\mathcal S$ the full subcategory category of $sSet$, of simplicial sets which have at most $\kappa$ many nondegenerate simplicies, all of which are elements in some fixed large enough "universe" set $\mathcal U$. $\mathcal U$ be chosen such that $\mathcal S$ contains also the generating cofibrations $\partial\Delta\left[n\right]\hookrightarrow\Delta\left[n\right]$ and the generating trivial cofibrations $\Lambda^{k}\left[n\right]\hookrightarrow\Delta\left[n\right]$ of $sSet$.

$\mathcal S$ is a small subcategory of $sSet$, hence cannot have all small (co)limits (because it's not a preorder). However, if one is willing to compromise the standard limits axiom with some weaker version of it, then perhaps $\mathcal S$ could be given a model category structure, one that is induced from the model structure on $sSet$? In particular, it would seem very appealing if $\mathcal S$ could be cofibrantly generated, with the same sets of generating (trivial) cofibrations.