No. Consider $X_0 = \{0,1\}$ with the metric inherited from $\mathbb{R}$. Let $X_i = \frac{1}{3}X_{i-1} \cup \left(\frac{1}{3}X_{i-1} + \frac{2}{3}\right)$ again with the metric inherited from $\mathbb{R}$. We are making the the standard Cantor set here with the metric inherited from $\mathbb{R}$. But each $X_i$ has dimension zero being a finite point set, but the standard middle-thirds Cantor set has positive box dimension. Needless to say that all these spaces are compact in the usual topology.
BSteinhurst
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