The answer is: Yes! Of course. That should not be a problem. But it may not come from your consumer level sources and more from other types of services. Here is what my experience has been.
If you are asking your web host, you may be wasting your time. There are web hosts that will allow you to do what you want, however, they are at another service level. Generally, if they are selling sites for $9.95, then they are not interested in much bother though not always. However, for companies such as Pepsi Cola and Coke Cola generally their sites are hosted at tier-1 sites and that can give you a start. You can research a few similar services and see where these sites sit by reversing the IP address.
Years ago, I worked on a temporary global web project that needed world-wide access and fail-over. I was able to find a host that not only could provide the systems, but the expertise on-site to handle all problems automatically all over the world. It was surprisingly affordable too! Not as scary as we originally thought though still moderately significant due to scale. The service level you want is available somewhere, you just have to dig deeper.
I used to be a web host and leased some rack space in a NOC along with a substantial block of IP addresses. I had 4 redundant OC3s, two to MAE-East and two to MAE-West which are the largest NAPs (network access points) for the Internet along with 3 satellite back-ups. At the time, there was absolutely no tier-1 provider with that level of access. I was basically bypassing any backbone.
I brought my own servers, firewalls, etc. If you are able to maintain your own systems, there is affordable NOC space you can lease, but you would not be renting this from a web host but from a service provider. You can go tier-1 such as AT&T or less such as your local teleco and any point in between. This is generally charged based upon plat-size (RUs rack units), power requirements (one plug or two for redundant power supplies), and network access speed. Network access speed can be unlimited or controlled with a bandwidth manager. Bandwidth managers do not have to be a bad thing. They can emulate bandwidth to a certain level that equates to a T1, DSL connection, etc. So you can start lower and easily dial-up your bandwidth with a phone call as you need it.
As well, you can also just get a commercial line and just run it at home without issue. When I was a web host, I had one or more commercial lines to my house along with a significant block of IP addresses and router options including fail-over and the ability to change upload and download speed on the fly (automatically) managed by my routers.
As well, I also leased space in a telco switch house, in fact, the largest in the world in Baltimore. I was able to place equipment there too. This allowed me to have an up-stream firewall on my commercial lines that easily defeated DOS attacks.
Of course this is perhaps more than you need, but you are getting the picture.
So while I cannot tell you immediately what rock to look under, I gave you an idea of what the rock looks like. Focus less on consumer based web hosts and research who some of the larger companies use. As well, if you are a bootstrapper (and I know that you are), you can always roll your own set of servers and run them at your house. With some level of embarrassment, I met the owner of the worlds largest and most profitable porn sites and he ran them out of his living room with full fail-over. He was an equal investor in a project that Sprint was also an investor. So I am not too embarrassed. I run my web servers out of my office today, though admittedly, far less sophisticated that years before. My teleco offers commercial connections for not much more than what I pay now with 80meg upload and download. I live up high in the mountains in a tiny town that is probably less than or about an 1/8th of a mile long so you should be able to do the same.