In a 5-to-6-quart slow cooker, place celery, carrots, onion, potatoes, and thyme.
Place corned beef (fat side up), on top of vegetables and sprinkle with the spice packet that comes with it.
Add water to cover most of the meat.
Cover and cook on high until corned beef is tender, 4 1/4 hours (or 8 1/2 hours on low).
Add the cabbage by placing it top of the corned beef. Re-cover. Cook until the cabbage is soft - about 45 minutes on high (1 1/2 hours on low).
If desired (and I do), remove whatever fat you can easily get to from the top of the corned beef.
Using 2 forks, separate the corned beef into pieces. Serve with the vegetables.