Thank you everyone for the many responses with wonderful explanations and suggestions. Our JewishGenners are always so helpful!
While my "cousin" and I knew it was possible that our connection was far enough back that we would not share any DNA, the most common thread among the responses was to check your DNA on numerous sites as sometimes you will match on one and not on another. With that in mind, I loaded my DNA to FamilyTreeDNA, a place where my "cousin" already had hers. Sure enough, here we find we are a match! 4th cousin to remote, but a match none the less!
We have been in touch for almost 30 years now, not knowing for sure if we were related. Yesterday was a day of miracles for us as we were able to finally find some proof of our connection. Chances are we may never find the actual connection, but it is nice to know that we truly are related.
Some of you asked if we cannot go farther back on our family lines. Sadly, not at this point. We have checked all the normal sources, but sometimes it is not possible to continue.
On my own line, I have gone back as far as my 3rd great grandparents, Mosek Izaak Klepfisz (b. 1940) and Esther Mandel. However, this is based on information from other's trees and I do not have specific records to verify the information.. My 2x great grandfather was Nison Mosiek Klepfisz b. ca. 1770. His wife was Szosa Ajzensprung. JewishGen does have a record for the marriage of their son Dawid Mordka Klepfisz to Etta Laja Ruda in Warsaw (whom I know to be my great grandparents), so this is my proof that they are my 2x great grandparents. There is also a record of death for Dawid, 4 Jul 1922 Warszawa, Mazowiedkie, Poland. As far as locations for my ancestors, I only know Warsaw.
As you can see, I continue to research and I do not give up. And in not giving up, my cousin and I finally at least have a shred of proof of our relationship.
Thanks again to everyone who offered suggestions. I am happy to report a bit of a success story!