Unfortunately the original question did not contain enough information for a clear answer, but then, as it was compounded with second a totally unrelated matter, I will attempt to clarify the second.
The problem is how the image is created and the format used.
Depending on the program used to create the image, there can be some extra information left on the RGB channels that should have not been part of the image in the first place. In other words, the alpha channel and the RGB information have not been associated. Saving as PNG only makes the problem persist, because it uses unassociated alpha.
If you look at the image in the UV/Image editor the alpha channel
You will can see the RGB information and you will notice that the image has those black bars, and some extra information that does not coincide with that of the alpha mask (6 extra elements on the bottom left of the image and some extra things on the bottom right as well).
The PNG file is using "unassociated" alpha, instead of "associated", where the pixels that are meant to be transparent would be multiplied by the alpha channel and should be black.
When you view the image in texture view, the information on the RGB channel is used. If you use any other view (material or render) the alpha channel will work correctly.
The proper way to deal with unassociated alpha is to multiply, the RGB with the alpha information.
Or to use it as a mask to mix the shader color or to mix different shaders.