I installed Point Linux on my laptop, and according to the battery settings, I have selected "Hibernate when battery is critically low". However, as the battery discharges, nothing happens until the power runs out and it just goes black. There are no settings on what constitutes a "critically low" battery.

I don't know if it matters, but I removed the "mate-screensaver" that came with it and installed xscreensaver. Xscreensaver has settings on suspending/hibernating after a certain amount of time has passed with no keyboard/mouse activity, but nothing on what to do based on battery power. That's handled by a different program.

(I know people will yell at me for using xscreensaver because it's not very secure, but I don't have anything sensitive on my computer; I just want screensavers for eye candy.) =P

Have you tried different graphics drivers. This is a guess only

Point Linux appears to be based on Debian... It would help to know what kind of hardware you are running on. For instance,... Asus laptops have their own quirky issues regarding battery readings. What is probably happening is that the battery will only report on each 10 of battery charge, and that critical is set to something like 5%... There has to be a config file for power management,... somewhere. I don't know enough about Mate (what Point Linux uses, apparently), to be able to tell you where to look for that file... But it has to be there...

It's a Dell Latitude 2120. And yes! It's Debian-based; I have installed a few *.deb packages. I get a popup warning when power drops to 10, but it doesn't turn off.

OK,... That system has an Intel Atom n550 processor (a 64-bit Atom), Max. 2GB RAM...

Hmmm,... Not too much about it, other that it runs, perhaps with bugs, on various older Ubuntus...

Try;

... at a terminal...

See if that gives you an interface to modify things...

It says that it's already installed. That must be the thing I'm using already.

One reason I like Point is that while having a nice comfortable interface, it's quite lightweight, using about half the system resources that Mint does. Good for a little laptop. ^^

I'd really recommend Xubuntu or the Mint XFCE equivalent... Precisely because there are so many users, so many packages for them, and more likelihood that someone else has your "quirky" box (not a dig on you,... I have some really exotic hardware too,... Like the original Dell Inspiron Duo 1090,... Which I believe also has the n550, if I remember right).

I don't think it's the hardware with the problem, because the hibernation sequence works fine when I boot into Windows.

Speaking of different distros, I'm trying to test out Haiku, but my ISO download always ends up corrupted for some reason.

It's not the hardware, per se... What it is, likely, is that the control over things like power management are not as well developed under Linux as they were under Windows for that device (MS has made a lot of effort to work with hardware manufacturers to tweak the OS installs for their devices with all kinds of custom power management cr@p,... I could tell you horror stories about how they've manipulated OEMs into putting troublesome hardware into things that only they knew how to fix).

That's why I'd recommend Xubuntu (14.04 LTS). It's lightweight enough to run on an Asus EEE PC 702, w/ 1GB RAM, 8GB SSD, and a 900Mhz Pentium Mobile processor... Plus, there's packages for manufacturer specific hardware junk, and the Intel Microcode stuff, as well,... Smaller Distros?!?! I just don't know...

That I can certainly believe. ><

I'll give Xubuntu a try sometime.

Also, what's a good OS-independent boot loader? I'm currently using Grub that came with my OS, but every time I toss in a new one, it takes over. I'd like something to run off of its own partition (or even just the boot sector) that gives me options to fiddle with. I read somewhere that there are even some that let you have more than one version of Windows installed on one HD.

Grub2 is it... Windows insists on installing its boot loader every time its installed, and fails to recognize anything but Windows,... Grub2 should be able to ID whatever OS you put on there... You might need to get comfortable with Grub commands if you are dual/multi booting... The last time I dual booted was Kubuntu/Scientific Linux on the same machine (the one I'm on now, as a matter of fact). The "trick" is restoring Grub (having scanned the partitions) after the subsequent install...

PS: Another thing to be aware of is that if you change your partitions, and your systems, say, share a swap partition (2 or more Linux installs), you will probably screw up the swap for the surviving Linux install afterwards (the volume IDs in the fstab file will reflect the old ones, and swap will fail). There's a way to ID the volumes and fix the fstab, but you have to be very careful.