In this case it means for something to be about to take place, so "half-crumbled yesterday," or something to that effect.
Other examples*:
父は若い時に、海で溺れかけたそうだ。
(I hear my father nearly drowned in the sea when he was younger.)
雨が降りかけたが、また天気になった。
(It started to rain, but it became fine again.)
飲みかけたビールのコップがテーブルの上にあった。
(There was a half-drunk glass of beer on the table.)
卒論を書きかけたんですが、まだ完成していません。
(I started writing my senior thesis, but I haven't finished it yet.)
*Examples taken from A Dictionary of Intermediate Japanese Grammar. I slightly altered some of the translations to better fit my taste, but the Japanese examples are Makino's and Tsutsui's.
かけた actually doesn't indicate the exact "half-way". It rather means to be gone or done some far. The idea also can be that something was about to happen but didn't happen or complete it. Either something actually happened and didn't complete or didn't happen at all depends on the sentence.
Let me give you examples
涙がこぼれかけた (tears did not drop nor dropped any but almost did)
死にかけた (did not die but almost did)
読みかけた本 (did read some but didn't complete)
It's obvious that something can not be done half-way. Just like a woman can't be pregnant only 50%. Either she is or not. Careful ね