There are some great answers already in this thread!
I've seen this question posted before and don't recall seeing any of the "big guns" reply.
1. The stem is damaged during the cutting/replanting stage e.g. use sharp scissors (no pinching) to cut the stems of larger diameter stem plants and dont crush the stem when you re-plant the top.
Plants are first damaged when cultivated for shipment.
LFS's handle them @ least twice before the third time when placed in a bag for your purchase.
Then it is up to you to unravel the bunch and determine what to trash from the bottoms.
I have heard that floating the plant and allowing it to develop roots on it's own, then planting it once there are roots there, will solve this problem.
Floating plants can work but who wants twisted and contorted stems that have been rolling around praying for light?
What I have observed is that if the conditions are such that stem plants will grow rapidly, there is no stem rot.
But, if anything is off so that they will not grow rapidly, you do get stem rot.
Like all generalizations, this one breaks down for certain plants for certain people.
Some stem plants refuse to grow for me no matter what I do. I solve that problem by growing other stem plants
Fortunately there are many, many to choose from.
Remember every plant is not for you.
Some don't like the soup you serve!
Here's the general idea behind what is referred to as "auto fragmentation".
Low nutrients, low CO2 particularly and low N, bad place to live.
Maybe current, maybe light etc, anyway, the plants break up in hopes of drifting away to a better place and regrowing,
sort of like seed dispersal, but much faster and better to find new and better habitat.
Generally poor CO2, poor nutrients(easy to rule out light and nutrients).
Tom Barr
Poor water quality in regards to growing plants. I didn't say phish.
I have noticed stem rot to occur on stems (mainly faster growers) because they get shaded near the bottom by new growth.
L.repens x 'cant remember the cross plant' is horrible and so is d.diandra too.
Some plants just do not like to be crowded.
This is also true of syngonanthis Madeira and tonina 'narrow leaf' in my tanks.
The stems have to be about one inch apart or after a week or so they will rot.
Shading & space requirements can play a factor too.
The only addition I would add is that I've seen high ammonia levels in a "hot" soil cause this issue.
One thing I've never seen is an instruction guide for people that are new to the hobby regarding "planting".