Larry and Anthony are both correct. You can tune just about any good condition drum both high & low, but whether it sounds good is (excluding tuning skills & head selection) quite dependent on shell construction, bearing edges, venting, size, etc. Also, it's not uncommon to find differences of opinion, primarily because the definition of "high" is so different from one player to another.
There are many variables here, but as context to help focus, essentially, a drum moves through several stages of "performance" in it's tuning range:
Starting with very loose, the earliest point at which the head first produces a discernible tone. At this stage, (with some exceptions) the shell typically isn't contributing much to the sound.
Raising up the tuning range, the point at which the shell starts to be excited by the head, & in turn, shapes the sound produced by the head.
Up a bit more, the first "sweet spot". The point at which the head is vibrating in the same frequency range as the shell. This is the point of maximum shell involvement, & often referred to as the "fundamental".
Up higher, there can be further "sweet spots", although these tend to be less wholesome due to overtone dominance.
Up higher, (again, with exceptions) we're now starting to fall outside of the head's ability to excite the shell around it's fundamental frequency range.
To the choke point, at which any shell tone is completely lost & the head note is so short as to produce little discernible tone.
The ability to excite the shell is often a key element in adding tonal depth to higher tunings, but not always. Very thick/low resonant shells are much more difficult to excite, but that doesn't necessarily limit their effective tuning range, just the resultant tone. Bearing edges have an influence too. A rounded edge shortens head sustain, & affects the result in two primary ways. It delivers greater direct vibration to the shell, & also reduces the build up of higher overtones, so is especially effective at offering a full tone at higher tunings where head sustain is naturally reduced anyhow. Conversely, sharp bearing edges on thinner shells deliver less into the shell. In combination with increased head sustain, this encourages a rather thin initial tone & dominance of higher overtones, often described as "boingy".
Then there's venting, & that dictates low & high dynamic performance more than anything, but is often perceived accordingly to dictate the choking point. An unvented shell will offer a more satisfying tone at low dynamic, even at high tunings, but will limit the tone available in medium & high dynamic playing, especially at higher tunings.
Wood species/shell construction: The fundamental tone's frequency & ability to excite the shell is very much dictated by this, & that in turn affects tones available at higher tunings. You may be surprised to know that, in shells that are easier to excite, a lower fundamental actually increases fullness at higher tunings. This is partially because the shell encourages low overtones to flesh out the resultant tone. At higher tunings, rerings have little affect other than offering the possibility of a thinner shell that stays round, although they do add to the mix at lower & mid tunings.
Head choice is a fairly important one here too. Often, a thicker single ply head will offer greater head sustain at higher tunings, & especially on smaller drums. Coated heads do a nice job of dampening higher overtones and feature the fundamental. Tuning choices, especially tuning both reso & batter head to the same pitch, will add a degree of head sustain at higher tunings, & is especially effective in combination with more rounded bearing edge profiles.
Hoops: Can open up or dry out a drum, essentially by adding varying degrees of mass primarily to the head, but also to the drum in it's entirety. Other than affecting head sustain, & to a lesser extent, shell sustain, their affect on high tuning ability is small. Key areas to consider are tuning accuracy, & adding mass to very thin shelled drums.
Believe it or not, this is a very basic summary of points. Simple principals, but huge numbers of variables. I hope this helps a bit, but I suspect it will generate more questions than answers.
