The paper places five different problems (thek-pebble game problem, two problems aboutk finite automata, the reachability problem for Petri nets withk tokens, and the teachability problem for graphs whose “k-dimensional” edge sets are described by Cartesian products ofk factors) into the hierarchyNLk of problems solvable by nondeterministic Turing machines ink-log2n space (and binary tape alphabet, to avoid tape “speed-up”). The results, when combined with the conjecture thatNLi contains problems that requireO(nk) deterministic time, show that these problems, while inP for every fixed value ofk, have polynomial deterministic time complexities with the degree of the polynomial growing linearly with the parameterk, and hence are, in this sense, “gradually intractable.”