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 -((ra)r)e-:
tabe-ru vs. tabe-(ra)re-ru
kak-u vs. kak-e-ru
However, the verb 分かる cannot. Why is that?
wakar-u
* wakar-e-ru
The verb 理解する, which has a same/similar meaning, can have the potential form. Suru verbs are known to be irregular and require -deki-, and 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-u
* sir-e-ru (Okay under the spontaneous or the passive interpretations of-((ra)r)e-, but not under the potential interpretation).
But for 知る, you can have a lengthier expression to turn it into a potential form. Nevertheless, 分かる even resist this form:
知ることができる
理解することができる
* 分かることができる
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).