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Looking at a part like this LM2596

It is rated for 3Amps with a voltage range from 3.3-12 volts.

Does that mean that this voltage regulator will only output a maximum of 3amps?

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That is an oversimplification.

From the data sheet http://www.ti.com/lit/ds/symlink/lm2596.pdf there are two relevant features, internal over-temperature protection and a maximum current limiter (Typical value 4.5A, minimum 3.6A).

The bottom line is that trying to get more than 3A continuous current from the regulator will probably not end well.

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