  In this article we analyze the post-Newtonian approximation of a
generalization of the symmetric teleparallel gravity with the help of the
parameterized post-Newtonian (PPN) formalism. This class of theories is based
on a free function of the five independent quadratic contractions of the
non-metricity tensor. By calculating the PPN metric of these theories, we can
restrict the Taylor coefficients of the free function with the help of the PPN
parameters and their observational bounds. We find two families of theories
whose PPN parameters are identical to those of general relativity, and thus in
full agreement with observations. For three further families, we find that only
the PPN parameters $\beta$ and $\gamma$ deviate, but can be brought arbitrarily
close to their general relativity values by an appropriate choice of the
Lagrangian, so that also these families contain candidate theories which agree
with observations. The remaining theories either possess no well-defined
solution of the post-Newtonian field equations, or possess a post-Newtonian
limit which exceeds the form assumed in the PPN formalism.
