Non-Legendary Planeswalkers
No, because even if it's not printed on the card, they're still Legendary due to errata:


  306.4. Previously, planeswalkers were subject to a “planeswalker uniqueness rule” that stopped a player from controlling two planeswalkers of the same planeswalker type. This rule has been removed and planeswalker cards printed before this change have received errata in the Oracle card reference to have the legendary supertype. Like other legendary permanents, they are subject to the “legend rule” (see rule 704.5j).


If you look at the Oracle text of Liliana, Death&#39;s Majesty, it says


  Types: Legendary Planeswalker — Liliana


instead of just "Planeswalker — Liliana" so it is legendary and you can only have one of her in play. But there could be newer planeswalkers printed in the future which aren't legendary, and of those you would be able to have multiple at the same time.

A few currently existing options to make non-legendary or at least multiple copies of planeswalkers (thanks to @AlexP and @ArcanistLupus for the suggestions):


Jace, Cunning Castaway
Helm of the Host for a planeswalker turned into a creature (e.g. with Memnarch and March of the Machines)
Mirror Gallery