Arduino 16 MHz with only 3.3V?
Officially, no. Practically, often but not always.
If I follow the datasheet to the the letter, this configuration should not work. BUT... After some test, it did work.
The data sheet does not say that it "should not" work, it says that it is that the part is not qualified for operation at 16 MHz below some voltage higher than you want to use. That does not mean that a given chip cannot work there, it only means that a chip that fails to may not be considered "bad". Failure may be total and obvious or subtle.
In practice I have seen it work, both on small modified import boards and those with switchable power supply voltage. But I also recall seeing it fail to work.
Two factors that can make a theoretical difference are part-to-part manufacturing variation and operating temperature.
Note that with an ISP setup you can change the fuses to run the chip on what was sold as a 5v external crystal board, to operate from the internal 8 MHz oscillator instead, and you can do this even while leaving the crystal on the board. If you also change the brownout detector fuse, you can lower the voltage further, for example to operate without a regulator directly on two AA cells. Of course that applies to the ATmega itself - any other active components on the board would have to be qualified for the intended voltage too.
A little test, not very scientific, but shows it runs fine on 3.3V - https://www.youtube.com/watch?v=If0eOa0Uj5I.
Again, if you would like to stay withing the specification just use a "clock divider/2" and run your MCU at 8MHz if they have a 16MHz Crystal:
clock_prescale_set(clock_div_2)
http://www.nongnu.org/avr-libc/user-manual/group__avr__power.html
According to the datasheet you need 3.78V to run at 16 MHz, and 3.3V isn't that far off. I would be surprised, for example, if it stopped working at 3.77V, because that would mean that the operating at 3.78V was marginal.
I have operated these chips out of spec, for example under voltage, or with a 24 MHz crystal. They seem to work. However I would guess that Atmel's position would be: if you operate out of our specs, don't come crying to us if it is unreliable.
A few percent out of spec will probably work - and 3.3V is 87% of 3.78V. They write those specs to allow for fluctuations in the manufacturing process, and are confident that all their chips would meet them. Probably a lot of chips would be over spec.
Including a way to use the internal clock (at 8MHz) of the ATmega328P would really help!
Set the fuses appropriately. See Engbedded Atmel AVR® Fuse Calculator
For the Atmega328P making the low fuse be 0xE2 should do it.
The reason I'm doing this is because 5V / 16MHz Pro Minis from China are way cheaper then 3.3 / 8MHz.
Be cautious that cheap clone boards may be manufactured with manufacturers' "seconds".