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NaI(Tl) scintillators can detect thermal neutrons with surprising efficiency, even in the presence of ambient gamma radiation. This is due to a peculiarity of the 128I level scheme. Deexcitation cascades following neutron captures in 127I, a main constituent of NaI with quite large neutron capture cross section, unexpectedly often involve the 167.4 keV state of 128I with 175 ns half-life, which feeds another isomeric level at 137.9 keV having a half-life of 845 ns. The 29.5 keV transition between both levels, as well as the subsequent deexcitation sequence to ground state comprising 137.9 keV sum energy, are almost certainly absorbed in the crystal. Signals including three scintillation pulses within few microseconds, the first one caused by a prompt part of the cascade, the second and third one disclosing 30 and 138 keV energy depositions and characteristic delays, are clearly related to such neutron captures. Double-pulse sequences comprising only the primary pulse and a delayed 138 keV signal are also distinctive but more prone to load-dependent background by random pulse pileups. This article demonstrates neutron detection with a Ø2” $\times2$ ” NaI(Tl) scintillation detector by finding and analyzing such double- and triple-pulse sequences in digitized detector signals. It quantifies effect and background rates as a function of the detector load ranging up to 30 kcps. Complementary measurements with a 6Li codoped Ø2” $\times2$ ” NaI(Tl) (NaIL) scintillator allow relating the neutron detection yield by delayed-coincidence counting to the 6Li capture rate in the same crystal and thus to evaluate prospects and limits of this technique. [ABSTRACT FROM AUTHOR]