Can any connected graph of $\ge k$ nodes be partitioned into $k$ components? Dyer and Frieze in "ON THE COMPLEXITY OF PARTITIONING INTO CONNECTED SUBGRAPHS" showed that the problem of deciding whether a planar graph has a connected-$k$-partition is NP-complete. On a graph with fewer than $k$ nodes, of course the partition is impossible. Cutting a graph with sufficient nodes into pieces strikes me as trivial. Is there a planar graph of $k$ or more nodes that can't be partitioned into $k$ subgraphs? When does Dyer and Frieze's problem result in a "no"?

You may have misunderstood the paper's definition of a connected-$k$-partition. It's on the first page of the paper. It doesn't involve partitioning into $k$ parts but partioning into parts each of size $k$. Moreover, it requires the induced subgraphs to be connected. An example of a graph on $4$ vertices that has no connected-$2$-partition is the star graph $S_3$.