The most prevalent criticism regarding budget carriers is poor customer service, which is logical (they have to cut all expenses).
On the first day, you may examine the server quality, but I feel two issues are critical: hardware failure and network outage. You can always restore a website from a remote backup, but you're not sure whether to wait for the server or give up and change the DNS pointing to a different location.
There are a few well-known budget providers, but many smaller companies (typically one-man armies) offer adequate servers. In general, I am a strong supporter of small enterprises.
1. How do you track down such little businesses?
2. How do you verify and trust them?