Is a 1965 quarter made of silver?
No. All circulating quarters dated 1965 and later are made of copper-nickel. The only silver quarters minted since 1965 were special coins struck for sale to collectors: 40% silver Bicentennial quarters minted in 1975 and 1976, and "Prestige" proofs minted 1992-present.
Yes, in 1965 some coins were made like that, in error, of course. A blank planchet from 1964 was stamped in 1965, resulting in a silver 1965. It's also possible that the coin was plated. Get a good scale; a silver quarter weighs about 6.25 gm while a plated copper-nickel quarter would weigh about 5.7 gm. FWIW, it wouldn't be a "misprint" though, it would be a "mis-strike". Bills are printed, coins are struck or…
It's not silver, just silver plated. That means it's only worth 25 cents. The last silver quarters were dated 1964. All circulating quarters dated 1965 and later are made of copper-nickel, not silver. The only silver quarters minted since 1965 have been made for special collectors' proof sets, and they were first made in 1992.
No, clad is used in the coin collecting community talking about the Copper and Nickel "sandwich" current, non-silver, coins have. So a clad quarter by definition would not have any silver because a clad quarter is a quarter with no silver. However, quarters were made of silver before 1965 so any quarter dated 1964 and earlier contains 90% silver, but those coins would not be considered "clad".
Easy Answer = all US Quarters bearing the date 1965 have no silver content. They're made of copper-nickel so they're only worth 25¢. Complicated Answer = To disburse the overflow of stored bullion silver, the US Treasury, specifically the US Mint, struck a number of silver coins - including quarters - diring 1965 & 1966 and disbursed them to banks - BUT these coins all were dated 1964.