Hierarchy for BPP vs derandomization
Let PTH be the hypothesis that there exists a probabilistic time hierarchy. Suppose the answer to your question is true, i.e., "PTH implies $BPP \subseteq TIME[2^{n^{c}}]$" for some fixed $c$. Then, $EXP \neq BPP$ would be unconditionally true. Consider two cases:


If PTH is false, then $EXP \neq BPP$. This is the contrapositive of what Lance noted.
If PTH is true, then "PTH implies $BPP \subseteq TIME[2^{n^{c}}]$" so again $EXP \neq BPP$.


In fact, even an infinitely-often derandomization of BPP under PTH would entail $EXP \neq BPP$ unconditionally. So whatever barriers apply to proving $EXP \neq BPP$, they apply to proving statements of the kind "PTH implies derandomization".