You could get Linux to boot very fast if you wanted it to.
Depends on the level of functionality you want. If you want a full desktop type thing, then no it won't boot fast. Thing is that it's very rare for a linux user to reboot, so until recently (with the rise of mobile devices), nobody gave a crap on how long it takes to boot.
Remember that the core of the OS is just the Linux kernel.. Linux is a kernel, not a OS. You can have a full fledged Linux distro like redhat for desktop use, you can use Linux to control mobile phones with very small amounts of ram, you can use Linux in a control box for traffic lights.
If you just want a Mp3 player, or a MAYBE a dvd player...
say you want a mp3 player
So say your using a computer that has good Linux ACPI support, you can turn on only the parts of the hardware you need.. I am guessing. Depends on how nice the componates are. Like you can control CPU speed and voltages on Intel CPU's if the chipset supports it.
So on the motherboard you setup a little 1-2 meg flash ROM with a simple Linux kernel, and a simple busybox based enviroment and a minimalist mp3 player. then you add a boot option to your BIOS so that a user can boot up into the on-board Linux rom.
On your harddrive you setup a special fat32 partition for the last couple gigs of the harddrive.
The minimal Linux install boots up and immediately mounts the fat32 partition and begins playing files from whatever m3u file it finds named "default.m3u". If it doesn't find the default.m3u file it does a quick search for any mp3 files and makes a default.m3u file of it's own.
If the user deletes the partition, it wouldn't work. But by making it like that a user can copy their mp3's to the programs to that special partition then it could.
Something simple stupid like that could be booted up and running in the time it takes you to pull your hand back from keyboard button that tells it to boot into the ROM instead of the disk. (something like the F2 button)
The more functionality you want, the more of the hardware you need to use, the longer it takes to boot up, the more comples you need to make the Linux OS.