Repeatedly finding and deleting maximal independent sets on a graph: Number of necessary iterations in restricted cases
A partition of a graph into independent set is the same as graph coloring. Therefore we can reformulate your question as follows:

Can we bound the chromatic number of a graph in terms of its maximum degree?

If every vertex has at most $c$ different neighbors, then the chromatic number is at most $c+1$, and this can be achieved using a greedy approach: scan the vertices in an arbitrary order, and color each vertex in a color different from all its already colored neighbors (this will always succeed since there are $c+1$ possible colors and only $c$ potential already colored neighbors).
There are graphs with maximum degree $c$ which have chromatic number $c+1$, for example the clique $K_{c+1}$ on $c+1$ vertices, so this bound is tight.