  We present lattice QCD calculations of nucleon electromagnetic form factors
using pion masses $m_\pi$ = 149, 202, and 254 MeV and an action with
clover-improved Wilson quarks coupled to smeared gauge fields, as used by the
Budapest-Marseille-Wuppertal collaboration. Particular attention is given to
removal of the effects of excited state contamination by calculation at three
source-sink separations and use of the summation and generalized
pencil-of-function methods. The combination of calculation at the nearly
physical mass $m_\pi$ = 149 MeV in a large spatial volume ($m_\pi L_s$ = 4.2)
and removal of excited state effects yields agreement with experiment for the
electric and magnetic form factors $G_E(Q^2)$ and $G_M(Q^2)$ up to $Q^2$ = 0.5
GeV$^2$.
