Enrico Fermi and neutron interactions Enrico Fermi and his team were studying neutron absorption and the subsequent gamma ray emissions, in the 1930’s era. They calculated, based on the size of the nucleus and the speed of the neutron, that the interaction time would be about 10^-21 seconds. What they found was gamma emission times were around 10^-16 seconds, much too long.  How did they measure this incredibly short time interval with 1930’s technology? 
 A: This observation was probably based (at least in part) on the energy-time version of the uncertainty principle, 
$$ \Delta E \Delta t \gtrsim \hbar/2,
$$
where $\hbar = 200\,\mathrm{MeV\,fm}/c = \frac23\times10^{-21}\,\rm MeV\,s$ is the reduced Planck constant.
If you have a population of unstable states which decay exponentially, $\dot N = N_0 e^{ -t/\tau}$, the lifetime $\tau$ is a good estimator of the uncertainty $\Delta t$ in the duration of any particular state.  In that case the energies of the excited states cannot all be the same $E$, but must follow some distribution with a range of energies $E \pm \Delta E$.  Usually the "width" of this distribution is referred to with an upper-case gamma, $\Gamma$, rather than as $\Delta E$.
Let's suppose you build a gamma-ray spectrometer and look for prompt gamma rays from some material under neutron irradiation. You'll observe a whole mess of gamma rays with energies from hundreds of kilo-eV up to a few mega-eV. Suppose you isolate a particular 100 keV transition and are able to confirm that all the gammas from that transition have the same energy within 1%, which is not an unreasonable demand to make on a spectrometer. Then the lifetime of the state emitting those gamma rays must be longer than
$$
\tau_\text{min} \approx \frac{\hbar/2}\Gamma = \frac{\frac13\times10^{-21}\,\rm MeV\,s}{1\,\rm keV} = \frac13\times10^{-18}\,\rm s
$$
Your naïve interaction time of $10^{-21}\,\rm s$ suggests "epithermal" neutrons, with energies of about 0.1 MeV.  For these neutrons, the existence of any spectral features at all in the MeV-scale photons emitted by the target nuclei suggests the neutrons are crisscrossing the nucleus several times rather than passing it by once at speed.
The two orders of magnitude from my $10^{-18}\rm\,s$ to your $10^{-16}\rm\,s$ were almost certainly not reached based on the intrinsic widths of the energy distributions, but on some other method which I don't know immediately. But the basic fact you refer to, that the neutrons seem to dwell in the nucleus for  "too long," can be seen from the existence of features in the photon energy spectrum.  This is a surprising case where shorter time intervals are almost easier to measure than longer ones.
