Abstract. Recently, Hundal has constructed a hyperplane $H$, a cone $K$, and a starting point $y_0$ in $\ell_2$ such that the sequence of alternating projections $\big((P_KP_H)^ny_0\big)_{n \in \NN}$ converges weakly to some point in $H \cap K$, but not in norm. We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago. Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (\`a la Genel and Lindenstrauss) and for the proximal point algorithm (\`a la G\"uler) are provided. We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor, Elfving, and Herman. Extensions to Banach spaces and the situation for the Hilbert ball are discussed as well.