Shortness of breath does occur as everything begins to squish upwards and therefore your lungs are also given less room.
I can't recall when mine began, but I know it was past 24 weeks. However I am sure in some women it does begin earlier. I think a lot is dependant on how your bump sits, your body and shape etc. My bump has always carried low so I only really noticed a bit later.
When you say you can't breathe exactly what kind of problem are you experiencing. Shortness of breath in PG women when their organs are squished up is fine as our bodies are designed to compensate for it and actually work fine, even though we ourselves feel we are not getting enough breath in. We in fact are. Its not a problem.
How are you lying at night? Sleeping on your left side is the optimal postion for circulation and breathing when PG (and any other time I guess) It also helps the placenta to work (when you are resting) at its best I read. Lying on your back may make you far more breathless.
I find not rushing up stairs etc helps. Also when standing up to take it slowly.
There can however be other reasons for it and if you are wheezy or getting really faint etc it may be best to nip in and see your GP just so s/he can have a listen, check BP, pulse etc for you.
Stuffy nose is also common when PG btw
I had one all through first tri and now again in third