My now 16 week old DD does this too. She has been doing it for about 6 weeks after previously taking at first breast then bottle feeds with no problems - in fact she used to guzzle her milk down. I have absolutely no idea why it started, and she was checked out by GP for sore throat/ear infection, but there are no problems. She is quite long and skinny but is still putting on weight (but only because I think I am just "perservering" with getting her to feed) but sometimes it is heartbreaking, sometimes frustrating and I have to work really hard to stay calm. She doesn't always do it - for example, if she wakes in the night, she takes her milk without any fuss - and the other day I had a whole day where she just took it, and drank every last drop from her bottle. I was amazed and started to worry that there was something wrong!
There are a couple of things that I do that seem to help from time to time - so in case you want to try them:
Sometimes - (weirdly) she will take the bottle if I start the feed standing up and holding her quite close to me - sometimes I have to sway or walk slowly around until she is happily drinking then I can gently sit down again
Sometimes - I will give her a dummy to calm her down then in my other hand get the bottle ready and make a swift switch
Sometimes - I have to sit her up and let her cry for a couple of minutes (this is tough especially in public as she wails like a small child rather than a baby and it sounds like I am trying to murder her rather than feed her), then she seems to take it fine
Sometimes - she will take it better if her head is slightly turned to one side - awkward for holding the bottle, but it works from time to time
I am hoping she grows out of it - so it's good to hear from others that this has been the case, and I am just hoping that I am not simply raising a child that doesn't like food and we'll have eating troubles for years to come (my younger sister was like this and my Mum said she used to do the same as a baby - she still doesn't "do" eating and is an American size zero unlike moi!)