Pretty much. My friends had a cheap Dell monitor that started doing that all of the time, as soon as you would turn it on, until you shut it off. It was only around three years old. My seven year old CRT started doing that a few months ago, but it does not continually whine, it comes and goes. That's the main reason I finally "updated" to an LCD. Funny thing is that I'm going to have to dig it out of the closet for about another week while I exchange my current LCD for a different one.
Funny thing about the whine too is that some people can't even hear it, while it drives others crazy.
Generally it means its on its way to the grave. But thats not always the case, at work we used to have this one CRT that whined ALL the time. I would walk over, and *THWACK* its side. Really hard. After trying to ram my fist through its side, it would shutup for a while. I was trying to break it, to get a replacement. The tech guy hates me..
Anyway, It lasted about 6 years. 6 FRIGGIN YEARS. It probabally still works, but all the monitors got replaced by LCD's. And I would beat on that thing about 2-3 times a week.