Many reasons.
The equation for bending stress is Mc/I, where M is the bending moment, c is the distance from the neutral axis of bending to the outermost surface of the material, and I is the area moment of inertia.
Because the diameter of wire that paperclips are made of is so small, the area moment of inertia (I) will be very small. Since this very small number is on the bottom of the fraction of the equation, the resulting internal stresses will no doubt be very high, eventually causing it to break.
You could fix this by using larger diameter wire, but then it would be heavy, and would not bend at all when you tried to paperclip a larger stack of papers together.
Another reason they fail due to fatigue is the cheap metal that paperclips are made of. Generally, the cheaper the material, the lower the yield strength.
A material will yield whenever the internal stress in a part (Mc/I) is greater than the material's yield strength.
Edited to add: You very well could design a paperclip with infinite fatigue life, but they would no longer be cheap, easy to produce, disposable, or economical.
I would rather forcus my efforts on something with a little more meaning