Can you activate a planeswalker loyalty ability and then activate another one on the same turn?

Each planeswalker has a special rule called the loyalty ability activation restriction: you may activate a loyalty ability only any time you could cast a sorcery, and you may activate no more than one loyalty ability of the same permanent per turn (CR 606.3). This means once you use one ability on a planeswalker, all other loyalty abilities on that same planeswalker are locked out for the rest of the turn.

The restriction is per-permanent, not per-player or per-turn in general. So if you control two different planeswalkers — say, Chandra, Torch of Defiance and Liliana of the Veil — you may activate one loyalty ability on each of them in the same turn, because they are separate permanents (CR 606.3).

Timing also matters: you can only activate a loyalty ability during your own main phase while the stack is empty (CR 606.3b, 307.1). You cannot activate loyalty abilities during combat, in response to spells, or on an opponent's turn under normal circumstances.

Example: You control Jace, the Mind Sculptor and you activate his +2 loyalty ability. Later that same turn, you cannot also activate his 0 or −1 ability, because you already activated one ability on that permanent this turn. The second activation attempt is simply illegal.

Some cards can grant exceptions — for instance, effects that say "you may activate loyalty abilities of planeswalkers any time you could cast an instant" change the timing but not the once-per-turn limit. Unless a card explicitly overrides CR 606.3, the one-ability-per-planeswalker-per-turn rule stands.