I tried Debian Wheezey and it kept crashing on my machine too. I have now switched to Linux Mint LMDE 201403 which is the Mint based on Debian testing. It's really stable and works like a dream.
Download it here, http://distrowatch.com/?newsid=08323

All the best!

This may or may not be relevant, but I'll post it anyway in case it is...

I had a similar problem with Windows 8 on my laptop. everything would freeze except the mouse, exactly how you described. Usually, If I walked away for 5 minutes or so, it would come back for a while. I also noticed that the hard disk was much more active than it should have been (confirmed by Task Manager reporting 90% + disk I/O and the hardware LED almost solid on). Long story short, Dell gave me a new HDD, but they replaced the mobo as well. Personally, I think just a HDD failure.

And yes, I know, Linux is not Windows (I have a link to the article in my sig) and Linux would probably respond to a HDD failure in a much more sensible way. I'm just reporting what I've experienced in the hopes that it can be useful.

So if you have a spare HDD laying around, it might be worth trying to use it and see if it makes any difference.

Yes, there is. Just type and it will display the manual for that command.

Thank you all, especially "selfprogrammed" and "maples" for info on the "man" command. I tried checking the syslog, nothing that seems interesting there, I'll try to check it when it crashes. I checked the dmesg too, but I'm not sure what to look for. Everything I saw regarding watchdogs there, seemed enabled.

Do note what I said about watchdogs. If you see them in dmesg, then you have some compiled in the kernel. They have been known to malfunction with some hardware and actually cause problems that sound similar to what you describe.
It may be that you need to remove that watchdog kernel code.
See the kernel documentation in /usr/src/linux/documentation/

If the watchdog code is in a module then that module can be prevented from loading by blacklisting it in /etc/modules.d.
However, they may be in the kernel core.

See if you can try a another version of Linux (one smaller and without such features as
watchdogs), just as a test. Someone may have an install CD that you can borrow long enough to
run Linux off the CDROM (you do not need to install it, just run it long enough to see if it fails the same way). Slackware comes with the huge kernel, and other simpler kernels, so maybe your distribution does too.

It crashed just a few minutes ago, complete with nothing happening at pressing crtl+alt+f1. I tried alt+prtscr+k and it worked, I got back to the login screen. I switched console (ctrl+alt+f1) and logged in, and after checking the man page for tail (thx again) I made it display 30 lines and I saw something suspicious. So I copied the log with cp (also copying the restrictions, d'oh! had to start gedit from root terminal) but this is the part that looked interesting, and a little bit around it.

Jul 16 19:35:05 debian gnome-session[4014]: DEBUG(+): GsmDBusClient: obj_path=/org/freedesktop/DBus interface=org.freedesktop.DBus method=NameOwnerChanged
Jul 16 19:35:05 debian gnome-session[4014]: DEBUG(+): GsmXsmpServer: accept_ice_connection()
Jul 16 19:35:05 debian gnome-session[4014]: DEBUG(+): GsmXsmpServer: auth_ice_connection()
Jul 16 19:35:05 debian gnome-session[4014]: DEBUG(+): GsmXSMPClient: Setting up new connection
Jul 16 19:35:05 debian gnome-session[4014]: DEBUG(+): GsmXSMPClient: New client '0x81bee0 []'
Jul 16 19:35:05 debian rsyslogd-2177: imuxsock begins to drop messages from pid 4014 due to rate-limiting
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: Gtk-WARNING: Overriding tab label for notebook
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: Gtk-WARNING: Overriding tab label for notebook
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: Gtk-WARNING: Overriding tab label for notebook
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: Gtk-WARNING: Overriding tab label for notebook
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: Gtk-WARNING: Overriding tab label for notebook
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: CRITICAL: get_column_number: assertion `i < gtk_tree_view_get_n_columns (treeview)' failed
Jul 16 19:35:06 debian gdm-simple-greeter[4047]: CRITICAL: get_column_number: assertion `i < gtk_tree_view_get_n_columns (treeview)' failed
Jul 16 19:35:13 debian gdm-simple-greeter[4047]: GLib-GIO-WARNING: Dropping signal ActiveSessionChanged of type (s) since the type from the expected interface is (o)
Jul 16 19:35:13 debian acpid: client 3985[0:0] has disconnected
Jul 16 19:36:08 debian gdm-simple-greeter[4047]: GLib-GIO-WARNING: Dropping signal ActiveSessionChanged of type (s) since the type from the expected interface is (o)
Jul 16 19:36:43 debian rsyslogd-2177: imuxsock lost 63 messages from pid 4014 due to rate-limiting
Jul 16 19:36:43 debian gnome-session[4014]: DEBUG(+): GsmDBusClient: obj_path=/org/gnome/SessionManager interface=org.gnome.SessionManager method=IsInhibited






The last two on 19:35:06. What is that?

I am also fighting with some nVidia drivers trying to find one that will not break something on my system. Several of the symptoms are the similar. I cannot get hand-off to console to work (CNTL-ALT-F1) getting several variations of unusable mess depending on which nVidia driver I am testing. I have tried 304.121, 304.88, 304.64, and am now using 96.14.20 which is not the correct driver for this card but has the advantage that hand-off to console works.

I do not know GNOME, as I use xfce4 now. I do note that stopping KDE often looks ungraceful (like a train wreak) as parts shutdown and other parts generate warning messages about it.
Especially so if you just kill some part of X and let it crumble to a stop.
You really have to compare that to some Gnome shutdown from a known situation.

However, someone is sending a whole lot of messages, (or someone stopped handling their messages). See what those messages were about would be more interesting.

You see the exact time and date of the events.
Try to grep that time in all the files in /var/log to see what else was happening around that time.
>> grep "19:36" /var/log/*

You can also combine greps to eliminate wrong stuff.
>> grep "Jul 19" | grep "19:36:3" | less

>> grep "Jul 19" /var/log/* > somefile
>> grep "19:36" somefile | less
>> grep "19:37" somefile | less

I'm sorry I can't keep sorting this out, I have now replaced the graphics card with the other one I was talking about. The problem with that was that I had to enable non-free in sources.list and install firmware-linux-nonfree to get it to work without Gnome going into fallback mode, but I think it works.

One reason I did this is that sometimes when I pushed alt+prtscr+K I got a message about "GPU lockup", so now I'm sure it's the graphics card and pretty sure it's hardware.

I will test this card for some time and come back and report if it has worked.

Sorry for late reply, I forgot about this thread. It has worked fine.