Clause learning is a technique used by backtracking-based propositional satisfiability solvers, where some clauses obtained by analysis of conflicts are added to the formula during backtracking. It has been observed empirically that clause learning does not significantly improve the performance of a solver when restricted to learning clauses of small width only. This experience is supported by lower bound theorems. It is shown that lower bounds on the runtime of width-restricted clause learning follow from lower bounds on the width of resolution proofs. This yields the first lower bounds on width-restricted clause learning for formulas in 3-CNF.