Hello everybody! I'm amazed how many pages are under this post, it seems like this particular problem is not so rare as it seems.
(I want to apologize beforehand for any mistakes, English is not my first language)

I, too, have this problem with breathing. It started 3 or 4 years ago when I've started working, and my job was really stressful for me. First 6 months I was so stressed that I was constantly tired, I was seeking out for long conversations with my friend and mother for support. A few times I even cried at work for intense fear and stress.
It was that time I've started this particular feeling every few breaths that I couldn't get enough air into my lungs, so to find relief I was trying to breathe in as deep as I could, sometimes it would work and I would find relief, sometimes it wouldn't. Sometimes it wasn't a deep breath, but yawning (this is really important! This symptom seems to happen to a lot of us). This state would plague me almost for a whole day, it was really exhausting. What was remarkable was that I was okay while sleeping. And what was really annoying when I woke up, was happy to breath easily and... a few breaths later this problem started anew!
Slowly, very slowly this condition passed, especially when I quitted the job and got into university again. But it seems that I'm not safe. At some times, really stressful, like times before exams, this state returns. For now it's not that severe as 3 years ago, but it's really annoying and, I'm sad to admit, it's in fact connected to anxiety. I don't have any particular problems with lungs or heart, but I have mild allergy (it seems to happen to a few of us, but it can be not connected) and atopic dermatitis... that flares up when I'm stressed... What it leads to... that I wanted to believe that I'm not an anxious person (I can compare myself with a friend who's really anxious, and I seemed really chill... compared to her...), but it seems that I totally am! (what can prove that I AM an anxious person is the fact that both my parents are also anxious, that have different reactions: my mother has irritable bowel syndrome and my father has coughing as reaction to stress). xD It's just that I don't experience emotions as intense or even mildly as other people, so I have difficulties with monitoring them, identifying them and properly managing them, so my low-key anxieties tend to manifest as physical aches or reactions, the most annoying of which is this hyperventilation without hyperventilation (I found this term in some article about physiology of hyperventilation).

I once asked my friend about it, who's a psychiatrist, and she just told me "it sounds like a panic attack". Just this panic attack is milder and longer. Really annoying.

I don't seek any answers or diagnoses, I'm pretty sure it's stress-induced. Just wanted to share, so that other people, who don't even know how to call their condition and how to search for that, don't feel as confused and alone in their quest to understand their condition.