I always print out on paper the key documents (flight and hotel bookings, etc) in addition to copying a full set of the docs/printouts to my phone and tablet as PDFs.

I've experienced many of the "easier to solve problems" mentioned by others already, additionally recently

• Rental car reservation paper showing better model booked than they tried to give me
• Rental car company not acknowledging prepaid fuel tank, quickly fixed by showing paper printout
• Rental car (again, why are most travel troubles with rental cars?) trying to upsell me to the level I had already booked and paid for
• Hotel claiming not prepaid, easy solution with hotels.com "already paid" booking printout
• Hotel claiming breakfast not included, when it was
• Paper boarding pass looks more convincing than something on my phone, when evicting seat poachers
• Accessing fast track security with booking paper at LHR when transiting and not yet having boarding pass for the next flight, as the security is before the transfer desks.

... and frankly just to keep myself in sync on complex trips where I stay in different hotel every night and fly somewhere each day ;-) I just keep the papers in time order and check what's next...

Bonus opinion: Even when I check-in online and have the boarding pass on my phone, I often stop at the kiosk to get a paper boarding pass. It's just that little bit more convenient at the lounge, the gate and on the plane, and allows me to double check my exit row seat allocation or change to a better seat if available or if the plane has been changed (many European airlines/LCCs seem to allow more seat selection at airport than online check-in ;-)

So, even if I have everything electronically, I find that paper still rules, for many reasons. Takes just some seconds for each trip to print the key documents and it has saved quite a lot of time and trouble.