If you want to go on a weekday, there's a good chance you won't have a problem but yes it is limited. If you want to go on a weekend, I've had reservations rejected even a month in advance for Muir in peak season on a weekend.
You can usually get a permit for Muir Snowfield which ends something like 1000 feet below Camp Muir, nobody wants to be there but it could be ideal for a training hike, might even be more pleasant than Muir since you wouldn't be with hundreds of climbers making noise all night.