Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.

If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A in Eisenbud).

However, if $A$ is infinitely-generated, then it is possible for the dimension of $A$ to be less than the transcendence degree of its fraction field. Take, for example, rational functions in one variable $A=k(x)$. Dimension 0, transcendence degree 1.

Is it always true that the dimension of $A$ is less than or equal to $n$?