  Relativistic bending in the vicinity of a massive body is characterized only
by the post-Newtonian parameter $\gamma$ within the standard parameterized
post-Newtonian formalism, which is unity in General Relativity. Aiming at
estimating this parameter, we use very long baseline interferometry (VLBI) to
measure the gravitational deflection of radio waves emitted by distant compact
radio sources, by Solar System bodies. We analyze geodetic VLBI observations
recorded since 1979. We compare estimates of $\gamma$ and errors obtained using
various analysis schemes including global estimations over several time spans
and with various Sun elongation cut-off angles, and analysis of radio source
coordinate time series. We arrive at the conclusion that the relativistic
parameter $\gamma$ cannot be estimated at better than $2\times10^{-4}$. The
main factor of limitation is the uncertainty in the determination of (global or
session-wise) radio source coordinates. A sum of various instrumental and
modeling errors and analysis strategy defects, that cannot be decorrelated and
corrected yet, is at the origin of the limitating noise.
