The problem may not be too serious. The solution could be as simple as a truss rod adjustment. Hopefully a proper set up by a trained guitar technician may be all you need.
From what you have described my best guess without looking at the guitar is that it has a back bow in the neck. If the 14th fret is where the neck joins the body, that is even more likely the case. The less likely possibility is a high 14th fret as Tim suggested.
The good news is, if your guitar has a truss rod, a back bow is not necessarily a sign of damage to the neck. If the guitar is a steel string guitar, it should have an adjustable truss rod (unless it is a really cheap guitar). The truss rod is designed to put counter tension on the neck to compensate for the tension exerted by the strings. If the truss rod is too tight for the gauge strings installed, it will overcompensate resulting in a back bow. This can be easily corrected with proper adjustment of the truss rod with the proper tool. To correct a back bow in the neck, you would loosen the truss rod. Don't try this yourself if you are not experienced as you can break the truss rod or strip the nut if you are not careful.
If your guitar has an adjustable truss rod, it will be accessed for adjustment either through the soundhole or by removing a small plate on the headstock between the tuning pegs. I say this not to suggest you try adjusting yourself, but to aid in verifying the existence of a truss rod on your guitar.
A trained or experienced guitar tech or luthier can inspect the guitar and let you know if it's worth performing a full setup. In a complete set up of an acoustic guitar, the truss rod may be adjusted, as well as the saddle height (the bar that the strings pass over at the bridge behind the soundhole). The adjustments will be made after a new set of strings are installed per recommendations of the expert and based on your preference for string type. Most guitars will allow for some flexibility in string selection without major modifications to the guitar.