Size-extensive Quadratic CI with single (S), double (D), triple (T), and quadruple (Q) excitations (QCISDTQ) and its non-iterative extension to a method correct at sixth-order perturbation theory, QCISDTQ(6), are developed and applied to electronic systems, for which full CI (FCI) results are available. It is shown that QCISDTQ results are more accurate than either QCISDT or CCSDT. In particular, QCISDTQ(6) correlation energies differ from FCI correlation energies on the average by just 0.04 mhartree. The improvement is caused preferentially by the balanced addition of connected and disconnected four-electron correlation effects, which confirms observations that four-electron correlation effects are important for a balanced description of electron correlation.