We did that (with Rackspace) for years before migrating to AWS. AWS is still far better from a service & flexibility perspective.
My employers website has certain times of the year where we see a huge increase in web traffic. When we had a hosted solution it took weeks of preparation to provision additional web servers to handle that load. We had to submit formal requests for additional servers, document how to wire them into our network & required firewall rules, etc. Then we had to wait an arbitrary number of days for them to do the work. And then we had to repeat that whole process when we no longer needed the additional capacity.
With AWS we just define an auto scaling group and additional web servers are spun up automatically when demand is high, and frees them up again when no longer needed. Even if we didn’t use auto scaling we could easily automate this sort of thing via terraform or other tools and spin up additional instances in minutes instead of days.