If you say “I’m up to my neck” in something, you're really busy and maybe even overwhelmed, like someone who’s making cookies for the entire sixth grade class in one afternoon.
Have you ever seen someone at the beach buried in sand all the way up to the neck? This expression is used for situations that feel like that, ones that are consuming or overwhelming. It’s not always bad. If you love knitting and you’re up to your neck in projects, that could be fun. But sometimes it just means you’re overwhelmed. A student with four hours of homework is up to her neck in homework.