Can nitrous dioxide develop when copper and dilute nitric acid react?
There are two ways to interpret an observation nitrogen dioxide.

Reactions between a metal and nitric acid rarely proceed by a single path to generate a specific nitric acid reduction product; in all likelihood there will be a combination of multiple reactions and thus a mixture of products.  With copper you could get both $\ce{NO}$ and $\ce{NO2}$.  And with a more powerful reducing metal like magnesium even more products could get into the mix, including ammonium nitrate.

You may have conducted the reaction in the presence of air.  $\ce{NO}$ is readily oxidized by the air to make $\ce{NO2}$ &quot;after the fact&quot;.  This is in fact part of the natural production of nitric acid in thunderstorms.  Further complicating this is the fact that nitrogen dioxide in turn reacts with water, forming nitric oxide and regenerating some of the nitric acid (this is also indicated in the above link).  To get the true product mix out of the nitric acid you needed to use an environment free of air and unintended water.