I recently asked this question Unbounded sectional curvature implies infinite diameter?. I would like now to ask something similar, but in another context.

Suppose you have a complete metric space $(M,d)$. Assume that the curvature of $M$ nas no upper bound. Can one concludes that the diameter of $M$ is infinite?

If it helps, one can assume that $(M,d)$ is the limit of a sequence of compact manifolds.