My guess is that the directory ~/bin/sbt/bin is not in your PATH.
To execute programs or scripts that are in the current directory you need to prefix the command with ./, as in:
./sbt
This is a security feature in linux, so to prevent overriding of system commands (and other programs) by a malicious party dropping a file in your home directory (for example). Imagine a script called 'ls' that emails your /etc/passwd file to 3rd party before executing the ls command... Or one that executes 'rm -rf .'...
That said, unless you need something specific from the latest source code, you're best off doing what paradigmatic said in his post, and install it from the Typesafe repository.