Why isn't the American nickel magnetic?
There are many types of magnetic properties, including ferromagnetism, paramagnetism, diamagnetism, antiferromagnetism, ferrimagnetism, superparamagnetism, metamagnetism, spin glasses, and helimagnetism.  Many of these are too weak to cause any noticeable interaction with a magnet. The type of everyday magnetism you're thinking of, which nickel has, is ferromagnetism.

While nickel is ferromagnetic,  copper is not. As you said, the American nickel is currently 25% nickel and 75% copper.  According to this paper (from 1931!), in order for a nickel-copper alloy to be ferrogmagnetic, it must contain at least 56% nickel:


  ... 56 percent nickel is required before the alloy shows ferromagnetic properties at ordinary temperatures.


https://journals.aps.org/pr/abstract/10.1103/PhysRev.38.828 

E. H. Williams, Magnetic Properties of Copper-Nickel Alloys. Phys. Rev. 38, 828 (1931).