I have no idea with the idea of MagSafe, but on my Retina MacBook Pro (also early 2013), it's become extremely finicky. About 20% of the time, it just doesn't charge. I just have to connect and disconnect it a few times and eventually it works.
Clean your connectors and the problem will be solved. Do you know what will be extremely finicky in a permanent way? Your Thunderbolt 3 port after it's had the power cable unceremoniously ripped out from it.
At least there are 4 of them, so when one breaks you'll be able to use one of the others.
Sometimes little bits of magnetic debris can get stuck in the computer side, e.g. iron filings, staples, etc. You can try to get these out with Blu-Tack or similar if the q-tip doesn’t cut it.
One time I noticed the connector was wobbly, and I found a very, very small piece of metal in there. I had hoped was the source of all my problems, but I think I caught it right when it happened because it didn't fix the problem.
Getting that metal out of there took a good half hour of trial and error. That magnet is powerful!
But maybe that's just the placebo effect.