  This paper discusses the effects of radiation damage to SiPMs on the
performances of plastic scintillator counters with series-connected SiPM
readout, focusing on timing measurements. The performances of a counter
composed of a $120 \times 40 \times5~\mathrm{mm}^3$ scintillator tile read out
by two sets of six SiPMs from AdvanSiD connected in series attached on the
short sides are presented, for different combinations of SiPMs at various
levels of irradiation. Firstly, six SiPMs were equally irradiated with
electrons from $^{90}$Sr sources up to a fluence of $\Phi_\mathrm{e^-}\approx 3
\times 10^{12}~\mathrm{cm}^{-2}$. The timing resolution of the counter
gradually deteriorated by the increase in dark current. The dark current and
the deterioration were reduced when the counter was cooled from 30$^\circ$C to
10$^\circ$C. Secondly, 33 SiPMs were irradiated with reactor neutrons. The
characteristics of counters read out by series-connected SiPMs with non-uniform
damage levels, were investigated. The signal pulse height, the time response,
and the timing resolution depend on the hit position in the counter, when
SiPMs' irradiation is not uniform.
