The issue seems limited to particular carriers, so it might have something to do with the specific equipment or software they are using for relaying text messages back and forth. For now, it affects a pretty diverse group of carriers, spanning from Vodafone Netherlands, through Mobistar Belgium, to India, and virtual operators connected to those networks as well.
The affected owners usually receive an error code 38, and yet there is no issue with actually receiving text messages. A Dutch tech blog has sent an inquiry to Google about the bug, and has received an answer that they are aware, and the text message sending cut is also present on Nexus 6. This goes to show that the carriers' interaction with Android 5.0 Lollipop is to blame, though we have no doubt that the brainiacs at Mountain View will find a workaround in no time.