I actually tend to disagree with this. If you have to sit on a server for a number of hours and are constantly keeping an eye on top or some other diagnostic output, you are reducing your overall productivity, and inhibiting you from pursuing other tasks (be it business or personal) at 100%. Even though you are not 'actively' doing work on the server for a large percentage of the time, I believe that the passive work involved is billable. In certain situations, you may be completely justified in billing for 4 hours of your time - if the work is outside of normal hours or if during these 4 hours your client is constantly messaging / calling you to check on the status of things, effectively making it impossible for you to carry on with any other productive tasks. In other situations, you might bill for less time, but this would be based largely on your relationship with your customer, your hourly rate, and your ability to conduct other revenue generating activities.
Additionally, I also believe that any other clients you have during this 4 hour period will suffer simply because you will always be distracted by this other issue. Let's say that you work 30 minutes initially on this issue, and then you monitor the box for 45 minutes. At this time, the box flips out and you have to spend another 10 minutes on it. Meanwhile, in the previous 45 minutes you started something for another client, and now his project is disrupted. If you're billing him by the hour as well, you're faced with having to keep track of many work stop / start times, which further reduces overall productivity.
I actually don't like to bill for projects by the hour because of these issues, particularly when high level technical skills are required for troubleshooting intermittent issues, server migrations, etc. I've found that these types of projects require a large time commitment in terms of general availability, but not necessarily for doing active work. Because of this we either charge clients a monthly retainer for a standard set of services (monitoring, maintenance, software upgrades, etc) and then we provide more time consuming services on a per project based fee schedule.