It is not possible for any web host to offer 'unlimited' bandwidth or diskspace. Web hosting plans like that always come with clauses that restrict how much bandwidth you can transfer at a given moment, how much CPU usage you can use, and so forth.
Most of the time, these hosting providers will offer plans like this but in the end, they will either shot you down in breach of their TOS, or try to upsell you to a hosting plan that will cover your bandwidth and/or CPU usage needs.