FYI, I measured a 1N4148.
I used Vd = n*VT*(LN 2) = 26mV
So I placed a 1N4148 in a closed box to shield from air currents and carefully wired a pico-ammeter to it making sure my cables & setup would not contribute leakage. When I sourced 26mV in the forward direction, I measured 2.6 nA (somewhat higher than I expected) at a stable temperature of 25C. This should be equal to the Is @ 25C.
A Spice model of a 1N4148 shows Is = 2.68 nA so the measurement seems feasible. Other diodes however may have Is many order of magnitudes lower so it will be more difficult to measure those small current. My electrometer can measure down to 1E-15 A with great pains but that's not a 5 minute test setup.