Can you hold a valid drivers license in Canada and the US at the same
time?  I have dual citizenship and residences in both places.

Just a free comment, and certainly no legal advice.

I believe there is no problem.  Dual citizenship is a bit
schizophrenic:  you are Canadian when using Canadian ID and related
documents; and US, when using US ones.  As one or the other, that is
your identity at the moment, a full citizen of that country with all
its rights, even if you happen to be in the other country.
As a resident of each country, you are probably even obligated to have
a driver's license from each, since in most places there are
restrictions on how long one can drive with a foreign or out-of-state
license.

But again:  no legal advice.

You can't officially "reside" in both places, like you can't reside in
two states.  One more thing, in both Canada and the US the driver
license is managed by the state or province, calling the dmv from both
places you live in would probably be the way to go.
I know in Quebec it's illegal to have 2 driver licenses.  You must
exchange your driver license to get a Quebec one.

I do not believe so. That said however, you can apply for, and receive
an international drivers permit which enables you to drive in many
different countries (including the USA and Canada).

http://www.voyage.gc.ca/main/before/faq/international_driving-en.asp

Should be able to answer most other questions you would have about it.

Reinedd and  Jtregunna,

I believe you both are right in the case of a person with only one
citizenship, but the questioner has both.  I think he can be a US
citizen with residence in the States, and as a Canadian citizen can
claim a primary residence in Canada,
a bit schizophrenic.  Neither country can deny him residency, nor
could the state or province deny him a passport if he applied for it. 
I rather doubt that a province can insist that he hand in a foreign
driver's license, only one from another province, ditto for a US
state.  The requirement is that after a certain period a resident have
a license from the state or providence in which he lives.

But as mentioned before, this is no professional or legal advice.

Myoarin