Yes, it's quite possible. Here's what I recommend.
You travel Antigua - Guatemala City - Flores - Belice - Chetumal - Cancun. Each section can be purchased separately, as I'm sure you will want to stay over in Flores (Tikal) and Belice. Here's Linea Dorada's site, which are luxury buses. http://tikalmayanworld.com/about/index.htm
Guatemala - Flores is about 8 hours and costs about US$ 30.00
Flores - Belice is about 4 hours (plus about an hour at the border crossing). US$ 20.00
Belice - Chetumal is about 3 hours, plus about 30 mins at the border crossing. US$ 5.00
Not much to see in Chetumal.
Chetumal - Cancun is about 6 hours. US$ 20.00
All routes are daily, with 8 buses daily from Chetumal to Cancun.
Much lower cost if you take chicken buses. If you do decide on this budget option, for the Flores - Belice sections, take a bus from Flores to Melchor de Mencos, walk across the border and then take a local bus from there to Belize City.
There are no chicken buses in Mexico.