I remember when I first built my first website (this was a looong time ago), the prospect of buying web hosting was a daunting one. I was still a relative novice in terms of programming (this was before Code Academy et al), so my website was pretty basic.
You can probably imagine therefore that the cheap web hosting packages sounded pretty attractive. Unlimited traffic, a huge amount of storage space, a load of free tools so you could easily install Wordpress or phpBB in a single click.
It didn't really matter that no one really visited the site or I didn't really know what half of the applications available to me really were.
In those early days, when traffic is minimal, it's probably reasonable for you to baulk at paying out for expensive hosting. Your cheap as chips package does the trick just fine.
You don't need to scale up an awful lot however for you to start to run into some problems with these packages. There's a good post over at Web Hosting Secrets Revealed that covers a lot of these issues.
The article mainly looks at the service offered by iPage, but the issues are fairly universal. For instance, that unlimited traffic that lured you in is really something of a honeytrap. Sure, they give you unlimited bandwidth, but you have very real limitations on CPU and RAM usage, both of which will limit the number of visits your site can get before you start running into problems.
The first sign you'll have of this looming issue are the emails you'll get from support staff telling you about the resource usage concerns regarding your account, and that as you're on a shared service, you should beware that you are impacting other sites on the server (oh, and have you thought about upgrading to our expensive premium package?).
Generally, the cheaper the host, the more sites are crammed onto the server alongside your own, so the lower the resources you have allocated to your own, and the greater the chance that your host will take your site down to stop it consuming any more CPU/RAM.
Depending on the quality of your host, when your site first goes down it can be really rather stressful getting it back up (that is of course, if they tell you your site is down in the first place).
Budget hosting tends to equate to budget customer support, and it's when things start going wrong that the few dollars you're spending per month really begin to bite. They'll generally lay the blame firmly at your door and won't admit that they've crammed far too many sites onto a single server.
It will be a case of 'fix' your site, or upgrade to something beefier, with all of the hassle that this often entails.
So whilst a super cheap host may appear attractive in those early days, you often save yourself a lot of hassle if you invest in something a bit better right away.