This question may be related to What does で分かる mean? (Cf. Tsuyoshi Ito and my comments to my answer).
(Regular) verbs can be turned into the potential form by attaching
tabe-ru vs. tabe-(ra)re-ru
kak-u vs. kak-e-ru
However, the verb
分かる cannot. Why is that?
理解する, which has a same/similar meaning, can have the potential form.
Suru verbs are known to be irregular and require
理解する does as well:
suru vs. dekiru
benkyou-suru vs. benkyou-dekiru
rikai-suru vs. rikai-dekiru
知る also does not seem to allow the potential form.
* sir-e-ru (Okay under the spontaneous or the passive interpretations of
-((ra)r)e-, but not under the potential interpretation).
知る, you can have a lengthier expression to turn it into a potential form. Nevertheless,
分かる even resist this form:
This seems to be indicating that the problem is not morphological or phonological (for example along the lines of crash with homonymous/homophoneme expressions that Flaw suggests).