This year I had to produce records from the previous four, including cancelled checks. When I went up to the bank web site to print them out, the on line records only went back a year. Everything else had to be ordered one-at-a-time through a cumbersome process, and they weren't free. Now I use a check register with built in "carbon" copies.
Also not great access to old records with my brokerage house, so I keep all my paper statements and transaction confirmations.
I have real problems with any electronic records stored at an off site institution. It's just too easy for them to decide not to make old records available, and when they do, you're SoL if you don't have your own copies. Call me old fashioned, but when all you "paperless" types are begging for records your bank will only produce at a significant cost to you, I'll be leafing through a file cabinet and will have what I need after only a few minutes. And before someone says "but what happens if your house burns down?" I've been doing it this way since the '80s, and nothing's burned down yet...
My bank sends an electronic statement. I toss it in my "finance" folder on my computer, where I can keep it forever and print it out anytime I want it.