You have to think that YouTube openly allows uploads and airs thousands of videos per day, from music to self-created. Not to mention there are scripts and programs to allow you to download the videos from YouTube, which further consumes their bandwidth.
A website dedicated specifically to video is going to run through bandwidth quite a bit quicker than a website with a few videos here and there, images and content.
It's always best to start small and work your way up the ladder. There's no reason to start with a dedicated server if the website hasn't launched just yet. Upgrading is easy enough to where when the times comes, you can simply transfer your data from one host to another via SSH (if you have quite a bit of data, otherwise FTP).
In regards to Disk Space and Bandwidth, there is no such thing as unlimited disk space or bandwidth. Period. End of Story. It's not been available for the past 20 years, it's not going to be available in the next 20 years :).
Hard Drives will always be limited to certain storage quotas, even when in an array or spread across multiple servers. Bandwidth will always be limited as it is one of the largest costs a web hosting provider has to incur.
The web hosting providers you refer to are typically shard web hosting providers, and there's nothing wrong with that, however, you won't be able to push 2,000GB (2TB) of bandwidth in a shared hosting environment. It's not going to happen, simply put.
To push 2TB of bandwidth on a shared server would not only bog the server down, but it would affect other clients, which the web hosting provider in questions isn't going to let happen, so you'll end up with your account suspended for excessive use.
These $2 to $6 a month hosting companies offering the world for nothing and making promises of unlimited disk space, unlimited bandwidth and high-resource offers are doing nothing more than marketing to your natural way of thinking..."Bigger is Better" when it's just the opposite.
You're going to be better off finding a reliable web hosting provider offering real, truly usable limits and paying a bit more. Trust me. If you value you're website at all and plan on doing much of anything with it, you'll need to invest, not go for the cheapest.
Typically, 1GB of Disk Space and about 10-15GB's of bandwidth from a truly reliable provider is going to run $5 to $10 a month. (Shared Hosting)