Creating Best Clusters of Objects Based on Distance Between Them
You can use Cluster Edge Deletion, however, the problem is NP-hard.

Create a graph where the vertex set is the set of your objects, and you create an edge between two objects if the distance is at most the threshold.

Now, the Cluster Edge Deletion problem is to find the set with fewest edges to delete such that the remaining graph is a cluster graph (a graph whose connected components are cliques).  In this solution, every object belongs to a cluster in which the distance to every other object is bounded by the threshold.

Alternatively, you can run $k$-means (with increasing $k$) until you're satisfied with a solution.  Take note that the $k$-means algorithm is randomized and gives an approximate solution (as also this problem is NP-hard), so you might want to run the algorithm several times per $k$.