A t-threshold scheme consists of \(s\geq 6\) pieces of information (shadows), such that 1) a secret datum X can be retrieved from any t of the s shadows; and 2) X cannot be determined from any t-1 or fewer of the s shadows. In this paper constructions are given of infinite classes of 2, 3 and 4-threshold schemes based on finite generalized quadrangles and projective planes. In each case there is a discussion of the security level when fewer than t of the shadows are known.