It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is.

Formally, the Shannon-Nyquist theorem states that if you sample a band limited signal at twice its bandwidth, an ideal reconstruction filter can be used to perfectly reconstruct the input signal. There's some wiggle room over ideal sampling/filtering, but the point is that it tells you exactly what the input was, provided it was band limited.

The misconception I think you're having is that band width is not the period of a signal.