Short answer: My guess is that there are Eddy currents in the copper, and dimes and quarters are a higher percent copper than nickels (because they have more nickel!).

Longer answer: The composition of US coins being minted now is some mixture of copper and nickel. This information is readily available online:

Quarter:    91.68% Cu  8.33% Ni
Dime:       91.68% Cu  8.33% Ni
Nickel:     75% Cu     25% Ni
Penny:      97.5% Zn, 2.5% Cu

Bet you didn't expect that- pennies are mostly zinc, and have the least copper out of all of them!

Anyway, only a few metals and their alloys are ferromagnetic - this means the metal has little domains which act like tiny magnets inside the material, which will align in the presence of an external magnetic field (like the field of a bar magnetic) and become magnetic.

Anyway, the only metals that have this property are iron, nickel, and cobalt (and a handful of their alloys). This paper studies the magnetic properties of copper-nickel alloys and reports, "although copper is only weakly diamagnetic it requires 0.8 or 0.9 percent nickel to neutralize this diamagnetic effect and 56 percent nickel is required before the alloy shows ferro-magnetic properties at ordinary temperature "

Which seems to indicate that the ratios of copper and nickel and American coins would not have magnetic properties at room temperature.

Coins struck at different times than today might have different compositions, which is why I wonder what year your coins were struck. For example, pennies, struck in 1943, were actually minted with steel in an effort to ration copper in the war.

Anyway, my guess is what you're seeing is the result of "Eddy Currents" in the copper. When you wave a magnetic around you're changing the magnetic field through the metal, which can induce electric currents. In a slab of copper, for instance, this can make little circular currents (kind of like little vortexes) which will in turn create their own magnetic field, pushing back on the original magnet. This video shows what happens when you drop a magnetic down a copper pipe- it goes slowly because of the force exerted on the magnet by the Eddy currents.

To test this, I recommend holding the coin at a steep angle, and if you have a small neodymium magnet, try to slide it down the coin like a ramp. If it goes slowly, it's probably Eddy currents in the copper.