The first thing to try/hope is that you have a bit of dirt in your SIMM sockets. It could be as simple as a bad connection. First check that all the SIMM mounting posts (the little arms at the end that have the tab that goes in the hole in the SIMM) are in good condition and not broken off. If any are broken off, you'll need to compensate for that when installing the SIMMs. For example, one can often put something in between the SIMMs which will push the SIMM in the broken socket into place.
Then, get some card stock, e.g. 3" X 5" cards. Fold one in half. Moisten the folded edge with isopropyl alcohol (91% if you can get it, but 70% will do). Use isopropyl alcohol without any addititives (no colors or moisturizers).
Slide the folded edge of the card in and out of the SIMM sockets to gently rub the SIMM socket contacts with alcohol. Move on to a new card as needed.
Let dry, try again.
Another test is to try a different set of four SIMMs, but it sounds like you've tried several different sets.
If you have another Mac which uses 30 pin SIMMs, test your sets of memory in that Mac. Try to eliminate the variable of bad RAM, or incompatible RAM.
If all of the above fails, you either have a failed component on the logic board which is affecting the memory controller/memory sockets, or your power supply may be at the ragged edge of functioning and extra RAM increases the power draw just enough to cause failure.
If you have a voltmeter, measure the voltage on the 5V rail and determine if it is close to 5V or somewhere closer to 4.9V.
If you have the power supply from a IIcx, IIci, IIvi, IIvx, Quadra 700, Centris 650, Q650 or PM7100, you can pull the IIsi logic board out of the case and power it up with the power supply from any of the above. Of course, a spare IIsi power supply would work as well and you would not need to pull the logic board out of the case.
It is also possible that leaking electrolytic capacitors have deposited corrosive goo on your logic board which has eaten through some trace essential to memory expansion. The stuff is hard to see. It looks a bit like long dried cola spilled on the board.