This seems to occur when RVM is used in a new user account, with a multi-user RVM installation (i.e. installed to /usr/local/rvm):
bash$ source /usr/local/rvm/scripts/rvm
$rvm_path (/usr/local/rvm/scripts/rvm) does not exist.rvm_is_a_shell_function: command not found
__rvm_teardown: command not found
The solution is to set the rvm_path variable in ~/.rvmrc :
bash$ cat ~/.rvmrc
This has to be done in order for the additions to .bash_profile or .bashrc to work.