The increased value of coupon payments is offset by the higher price, or premium, that an investor would have to fork over for the bond. Surprisingly, bond prices don't act the same in both interest-rate directions. The change in a bond's price on the upside, when rates fall a set amount, is greater than the downside change, when rates increase by the same amount. In fact, as interest rates rise higher and higher, the decline in a bond's price gets lower and lower. The law of diminishing returns is a nice analogy to this, but there's a twist, here it is the law of diminishing declines.
Time is another major factor, besides interest rates, that affects how a bond's price changes.
Bonds with longer maturities are more sensitive to changes in interest rates than those with shorter maturities. For example, if market rates rise above the rate that you as a bondholder are receiving, you will take a loss because you could be earning more from a bond that is issued today. The longer you hold on to your lower-yielding bond, the larger your loss becomes. Maturity is not an accurate measure of how sensitive a bond is to changes in interest rates because different bonds have different payment schedules. So, the higher the coupon is, the shorter the effective maturity.