Two United States coins that add up to 35 cents are one quarter and one dime. One of these coins is not a quarter, satisfying the key condition stipulated in the question.
As of 2014, a variety of denominations of United States coins are in circulation. Consider the four coins that are worth less than 35 cents, which are the penny, nickel, dime and quarter. Half-dollar and dollar coins are eliminated from consideration, as they are each worth more than 35 cents. Two dimes are worth a total of 20 cents, which is not enough.
One might become confused by the wording of the question and assume that neither coin can be a quarter, but this is not the case. The parameters of this problem are that two coins are required, the total value must be 35 cents and that one of the two coins cannot be a quarter.