by my understanding, as a native english speaker learning chinese: 能 = possible. 不能 is like "there is no way that will work, it's not possible, that cannot be done" 行 = to run something, to conduct something, to go underway, therefore 不行 = that won't do, that's won't work.

That is an interesting question. I think, 不能 is often used when there is a verb in the sentence, e.g. 他不能说英语. 不行 is seldom used with a verb. Usally 不行 is used as an anwer to refuse some request. For example, “妈妈，我可以玩游戏吗？“ We can answer ”不行“ or "不可以"

1.in terms of meaning: 不能 means "can not" or "is not able to " e.g.我不能和你一起去北京。 e.g.这是他的秘密，我不能告诉你。 不行has two meaning.one is "can not or to be allowed" the other " be good at". e.g.明天和你去北京？我不行。(Go to Beijing with you tomorrow? No,I can't.) e.g.你想请假？不行! (you want to leave? No, you're not allowed) e.g.你觉得我中文不错？我不行,我只会一点儿。(You think my Chinese is not bad? No,I am not good at Chinese,I just know a bit. ) 2.in terms of Chinese structure 不能 as an auxiliary verb. have to be followed by verb and phrase of verb.But 不行 can not. and it is used alone. e.g.A:下个星期，你能陪我去看电影吗？ B:不行，下个星期我很忙。（or 下星期我很忙，不能陪你去看电影） 3.a set phrase of 不行 is 不行了。 不行了 means can not do something anymore. e.g.我不行了，我吃得太多了。（I have eaten too much,I can't eat anymore.） e.g.太累了，我不行了。(I'm tired very much,I can't do it anymore)

"不能 ”means you are not allowed to do sth or you haven't the ablility to complete sth,but for "不行 "we answer only with it ,for example: A:那张报纸能给我看吗？ B:不行！我正在看。 A:你明天有空和我一起去吗？ B：不行啊，我明天没空不能和你一起去。 I believe you can get the difference!^^ :

“不能”语气较为强硬，几乎等于“禁止”。“不行”语气要缓和些 “不能”：你不能在车里吸烟！ 你不能逃课！ “不行”：这么做是不行的