Nop. I think there are three common colloquial usages for 可:
可+adjective+了: stresses the adjective. 可 means "so, very". E.g. 可孤独可寂寞了.
可+verb+了: 可 means "finally; at last; I've been waiting for that". E.g. 他可走了.
可别/可不要: stresses "don't". E.g. 你可别走.
I think it just means "so alone, so lonely" (can't really explain the difference between 孤独 and 寂寞).
可……可…… is not a sentence structure, but two different statements, in fact you could also write that 可孤独、可寂寞了 (although there's often no pause in the speech).
Btw, the correct ellipsis in Chinese is "……", which you get by pressing shift+6 in most Pinyin input methods (I don't know why the font on my phone displays them at the bottom. I think they are supposed to be mid-line according to the Mainland/Simplified Chinese standard).
Although a lot of people on the Internet type 。。。 like you did cause they don't know how to use their IME.