Session Manager start-session hangs when root partition full

I dunno if this is the appropriate place for this, but when I attempt to start a session (either via the AWS Console or the CLI--via the session-manager-plugin) onto an EC2 will a full root partition, it just hangs.
In the console I get a blank screen for a long time and then eventually a blinking cursor that doesn't work.

In the CLI (via session-manager-plugin), I get a message that it's starting a session but then it just hangs.
The CLI/plugin doesn't respond to Ctrl+C or Ctrl+D; in fact, I have to start a new terminal on my workstation and kill the CLI command and the plugin command.

Start by reproducing the hang with a full root partition on an EC2 instance, using both the AWS Console and the CLI through session-manager-plugin. Trace where session startup stops and verify that the console and CLI either complete or return a usable error, including that the command can be interrupted normally.

Written by the indexing model from the issue text.