Answer: What you describe sounds like crown rot which is common in begonias when they are set too deeply in potting soil or when potting soil remains too wet. This fungal disease attacks the vascular system and essentially clogs the transportation of fluids from the roots to the stems. That is why the stems look fine one day and collapse the next. I don't know whether you can save your plant, but you can try. Begin by unpotting it and using a knife to cut away any mushy or browning parts of the crown and the roots of the plant. Disinfect the knife between cuts (use 1 part household bleach diluted in 9 parts clean water). After removing any diseased material repot your begonia in fresh potting soil, setting the crown of the plant just barely below the soil surface. When you water your begonia, water from the bottom (set the pot in a trayful of water) so you don't splash water on the stems. Allow the top half-inch of soil to dry out before watering again.
Hope this helps!
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