(Kanninbukuro no o ga kireru; “The string on one’s bag of patience breaks”)
Someone reaches the end of their patience and explodes. Running out of one’s ability to hold back and put up with something, especially after suppressing anger for a long time. The image is of one’s endurance being a sack that must contain one’s anger, which spills out when the draw-string suddenly snaps.
We begin with the a compound noun comprising 堪忍 (kannin), “patience,” and 袋 (fukuro), “bag.” The particle の (no), here in its role as a possessive marker, connects the bag to its 緒 (o), “cord,” which in turn is marked by the particle が (ga) as the subject of a verb. And that verb is 切れる (kireru), “to break,” “to be cut,” in sentence-final form.
The final particle and verb may be replaced at times with を切らす (wo kirasu), “to run out of.”
Apparently this phrase comes to us from the 堀川百首題狂歌集 (Horikawa hyakushudai kyouka shuu), an Edo-era collection of satirical poetry.
(“Tsui ni kanninbukuro no o ga kireta no ka, sensei wa suugaku no rekuchaa wo akiramete nijuppun hodo gakusei tachi wo shikatta.”)
[“Maybe his store of patience had run out; the teacher gave up on the math lecture and spent about twenty minutes telling the students off.”]