You need to specify the whole path:
tar -xf playground.tar playground/test.txt
You can use tar --list or tar -t to list the contents of an archive to see what's in there:
$ tar -tf playground.tar
playground/
playground/text.txt
Here's a complete log of what I did to reproduce your issue:
$ cd $(mktemp -d) # Go to a new empty directory
$ mkdir playground
$ touch playground/test.txt # Make the file we will tar
$ tar cf playground.tar playground # Make the tar
$ tar -tf playground.tar # List the contents of a tar
playground/
playground/test.txt # There's our file! It has a full path
$ rm -r playground # Let's delete the source so we can test extraction
$ tar -xf playground.tar playground/test.txt # Extract that file
$ find . # Check if the file is now there
.
./playground.tar
./playground
./playground/text.txt # Here it is!