The liveness of concurrent objects despite asynchrony and failures is a fundamental problem. To that end several progress conditions have been proposed. Wait-freedom is the strongest of these conditions: it states that any object operation must terminate if the invoking process does not crash. Obstruction-freedom is a weaker progress condition as it requires progress only when a process executes in isolation for a long enough period. This paper explores progress conditions in n-process asynchronous read/write systems enriched with base objects with consensus number x, 1 < x n (i.e., objects that wait-free solve consensus in a set of x processes). It is easy to solve consensus in such a system if progress is required only when one of the x processes allowed to access the underlying consensus object invokes this object and does not crash. This paper proposes and investigates a stronger progress condition that we call x-wait-freedom (n-wait-freedom is wait-freedom). While it does not need more assumptions than the previous one in order to ensure progress, that condition identifies additional scenarios in which progress is required despite the fact that none of the x processes allowed to access the underlying consensus object participates. The paper then presents and proves correct a consensus algorithm that satisfies this progress condition.