My short answer is...it depends. For contingency, I would. For normal diving, no.
If I get somewhere and my computer craps out I would have no problem doing so as I always dive with an additional time keeping device (watch) and a depth gauge and carry tables with my diving equipment. Having said that, I would ensure that I plan and execute my dive(s) accordingly..ie plan the dive using the tables (to give me the numbers) and then be a cling-on to my diving partner (not so with Insta-buddies though). BTW, this works if you are on the same diving schedule (like a diving trip somewhere) and dive with the same partner.
I always carry dive tables in various shape or forms which I would use and rely on these numbers for dive planning. I could rely on these for the dive should I become separated from my diving buddy. Recreational dive...that should be no problem whatsoever as if we were within NDL before separation and we follow the procedure I should still be within NDL when I surface as I or my regular diving buddies never attempt to stretch our dive into the last minute of NDL before coming up.
More advanced dives requiring deco...I dive with two computers (one in gauge mode) and a watch. If my main one was to fail, I would still have my two independent time pieces and redundant depth gauge. The profiles are normally square and based on tables and timings. Here again, follow the dive profile and the planned deco schedule. If in doubt, dive the more conservative profile in the team.
The thing to remember is the computer is a tool and not a crutch. It is there to assist me and not something I depend on. When I started diving, technology was far from being what it is today. I dove with a horse collar, tank back pack, no SPG (good old J valve) and no computer.
Do I use technology...sure. Can I still practice this hobby of mine if some of those tools craps out...you bet.