I've looked a few times in different sections of the tiny84 and mega328P datasheets, but I can't find an explicit answer to this.
Are there voltage/frequency limits for the Low Power Crystal Oscillator, or is it capable of running at any frequency and voltage within the recommended operating conditions? The Full Swing Oscillator has a note in its section (helpfully buried at the end of a paragraph instead of in a proper table) that says it only works above 2.7V, which the Low Power one doesn't have. The assumption would be that the Low Power one doesn't have that limitation, but I don't like making that assumption. That's the kind of thing that leads to unpredictable and inconsistent behavior, the worst to debug.
The scenario I'm thinking of would be if I wanted better accuracy than the internal RC oscillator, I've got a lot of 16 MHz crystals. Instead of buying a lower frequency one, hook that to the micro and program the CKDIV8 fuse so it starts up at 2 MHz by default. That way it can run all the way down to 1.8V.
It is not very clear from the data sheet ( http://www.atmel.com/images/Atmel-8271-8-bit-AVR-Microcontroller-ATmega48A-48PA-88A-88PA-168A-168PA-328-328P_datasheet_Complete.pdf ) that I looked at either. Near the top, they talk about the frequency range and mention the "speed grade" 0 - 4MHz@1.8 - 5.5V, 0 - 10MHz@2.7 - 5.5.V, 0 - 20MHz @ 4.5 - 5.5V which seems to indicate it is a static device and can go all the way down to 0 on the frequency, however, in looking at the area around pages 29 and 30 in the data sheet above, it seems to be more along the lines of 400khz as the lower end of what they are showing. You might try contacting support at Atmel and asking them - some processors I have worked with in the past are "dynamic" and have a minimum frequency to be able to "remember" what they are doing while others actually are static and can go all the way to 0 on the clock without "forgetting" what they were doing. You may also find a way to do what you want in looking at the sleep modes since they affect what the clock is doing.