I love this as a nickname for Elisabeth (that being my online pseudonym). I know it may be rather odd and/or controversial but it allows me to create another 'person' or identity for myself. Not to catfish people or anything like that; just for me. 'Tetty' is a lovely and unusual nickname for the beautiful 'Elisabeth'. Even though it isn't my name per se, I still feel a personal connection to as it is the name I chose for myself (Elisabeth 'Tetty' Jordan) and it allows me to be a different person every now and again. It's almost like a child playing dress up or pretend. Not to con or catfish but for fun and the chance to be someone you're not.
Elisabeth Johnson was the wife of famous Enlightenment writer Samuel Johnson, and he often called her "Tetty" or "Tetsey," as those were common (if rather girlish) diminutives at the time. James Boswell, Johnson's good friend, affectionately razzed him about it a bit when he did his biography, "The Life of Johnson."
Out of all possible nicknames you can get out of Elizabeth, this is probably the worst one. It's worse than Buffy! In American pronunciation, it sounds just like Teddy. In British pronunciation, it sounds a bit like 'titty'. And it looks like it in both British and American. This is an ugly and silly name. It's quite infantile as well.