Assumptions/context: Let's say we have some objects $X_i$ in some category, with $i$ belonging to some index set $\mathbf{I}$, such that there exists some object, call it $X_{\mathbf{I}}$, in the category satisfying the universal property of the product for the $X_i$, $i \in \mathbf{I}$.
Question: Do any counterexamples exist of strict index subsets $I \subsetneq \mathbf{I}$ such that no object in the category exists satisfying the universal property of the product for the $X_i$, $i \in I$?
Rephrasing of question: Does the existence of an object $X_{\mathbf{I}}$ satisfying the universal property of the product for the $X_i$, $i \in \mathbf{I}$ imply that, for all $I \subsetneq \mathbf{I}$, there also exists some other object $X_I$ satisfying the universal property of the product for the $X_i$, $i \in I$?
Notes/Comments: If $\mathbf{I}$ should be an "index category" instead of an "index set", and $I$ a "strict index subcategory" instead of a "strict index subset", then fine. I am less familiar with using categories to index things though.
An example would be a category where finite products do not exist but infinite products do exist, for some reason.
Comparing with the question title, the "object $X_{\mathbf{I}}$ satisfying the universal property of the product for the $X_i$, $i \in \mathbf{I}$" is "the categorical product of the family", whereas the "object $X_I$ satisfying the universal property of the product of the $X_i$, $i \in I$" is the "categorical product of the subfamily".
The empty product is (if it exists) the terminal object. Now, I construct a category with at least one product of two objects but no terminal object.
It is simply the category $X\leftarrow P \rightarrow Y$. Note that $P = X \times Y$ in this category but there is no terminal object since for every object $c$ there is at least one object $d$ so that $\text{Hom}(d,c)$ is empty.
Proving that the limit of the empty diagram $\emptyset \rightarrow \mathcal C$ is the terminal object is a nice exercise on the definition of the limit.