Identifying wordlike units in language is typically done by applying a battery of criteria, though how to weight these criteria with respect to one another is currently unknown. We address this question by investigating whether certain criteria are also used as cues for learning an artificial language--if they are, then perhaps they can be relied on more as trustworthy top-down diagnostics. The two criteria for grammatical wordhood that we consider are a unit's free mobility and its internal immutability. These criteria also map to two cognitive mechanisms that could underlie successful statistical learning: learners might orient themselves around the low transitional probabilities at unit boundaries, or they might seek chunks with high internal transitional probabilities. We find that each criterion has its own facilitatory effect, and learning is best where they both align. This supports the battery-of-criteria approach to diagnosing wordhood, and also suggests that the mechanism behind statistical learning may not be a question of either/or; perhaps the two mechanisms do not compete, but mutually reinforce one another.