Have you been receiving reports from monitoring services telling you that your site has been down for a significant amount of time? When you queried this with your hosting provider did they respond that the server had been up 100% of the time?
Here you have run into a situation where you and the hosting provider are talking about two completely different things. You are talking about how your site has been down for a period of time while they are saying that the server hosting your site has not been down. Both of these can be correct at the same time. The server may have been up all the time but that doesn't mean that your site has been accessible all of that time.
Communications on the web use what are known as PORTs to distinguish the different types of communication. The server listens on a number of different ports for various types of requests so that it can respond to them. The HTTP protocol that people use to access your web site uses port 80 so when someone tries to access your web page a request for the page arrives via port 80. Provided that the web server is running at the time and the port isn't blocked by a firewall then the server will receive the request and respond by delivering the page.
Just because the server itself is on doesn't mean that the web server software (eg Apache, IIS etc) are running. Even if the software is running it doesn't necessarily mean that the port is actually open and that the server is able to receive and respond to the request. Most servers have a firewall in operation and while the firewall is supposed to be configured to not interrupt requests on specific ports (port 80 in particular), if the firewall did block that port for whatever reason then your site would be inaccessible even with the web server software running.
Now I am not suggesting that any of these are necessarily the cause of your site being down. The point is that it is quite possible for the server to be on and appear to be running normally in so far as the hosting provider's monitoring is concerned and still have your site down for extended periods.
The problem in these instances is in getting the support people at your hosting provider to even acknowledge that your site has been down. Their immediate response to your complaints about downtime is to simply provide you with a copy of their stats that shows that the server was on throughout the period in question.
So is there a way to tell that the server itself is up even though your site is down? Yes there is. Only web pages care communicated via HTTP on port 80. Your hosting provider will generally give you several other ways to access your hosting - FTP, cPanel, webmail etc. - and each of these operates via a different PORT. If you can access your hosting via FTP or cPanel while the actual web pages are inaccessible then it means that the actual server itself is still operational and that the problem lies elsewhere. It doesn't mean that your site isn't down, it just means that convincing your hosting support that your site is down is going to be more difficult as you will need to go beyond their initial response.
As an example I had an external monitoring service advise me that my site had 99.74% uptime for a specific month. The times it reported the site as down included one time when I was actually trying to access the site both online and via FTP. The FTP updates worked but I still couldn't access the site online. My initial query to support brought the response that the server had been up 100% of the time. Only after a much longer discussion did they respond that the web had only been available for 99.709% of the time and that they were going to make changes to the configuration to try to ensure better uptime.