Many of these radios (like the AOR8600, R8500, ect.) advertise pretty much a DC to daylight coverage, with the exception of the ECPA-banned ranges in the 800 band to 'protect cell phone' users. Generally speaking, that means either 500khz (right at the bottom end of the MW spectrum) right up into the Gigahertz ranges (take a look at the AOR, Yaesu and Icom wiki pages...)
However, how well you will hear across that large a range has much to do with the design of the receiver - you'll notice I didn't call them scanners - and what antenna(s) are used on them. (Where you are in the world also plays a part...I'll explain that in a moment). The kilobuck receivers will generally do a pretty good job across the board - The Icom R8500 is a real popular choice, tho there are other older models that also perform reasonably well.
This depends to some extent on where you are in the world. On the HF side, signals tend to be stronger on the East Coast (since many broadcasters beam this direction), and propagation from European stations is a bit easier. Europeans have the distinct disadvantage of having so many broadcasters around them that you could make a case for them being in a saturated RF zone.
This is where the big receivers like the 8500 do a much better job than their cheaper cousins such as the AOR8600 or even a handheld like the Icom R20. Put a good long wire on them in these areas, and at night, overloading becomes a real hassle; the R8500 (and others like it) most likely won't have any problems at all. Too, they may not be able to handle seperating a weaker station amongst stronger ones (altho some, like the 8600, can have seperate tighter filters installed - that's only half the story...).
Therefore, your best bet is to get a good HF radio - there are many out there, our Wiki pages have several of these, but there are still more we don't have any on - and a good scanner as a seperate unit. You will be better off in the long run.
73s Mike