Nightveil Specter simply lets you play the cards it exiles, it doesn't give you any other special permissions. You're still bound by the one-land-per-turn rule if it exiles a land, just as you would only be able to cast a sorcery with it if it was your main phase with an empty stack, etc. etc.
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That said, since it says "You may play cards exiled with Nightveil Specter." as a separate ability from the exile effect, you will be able to play the land you exiled on your next turn instead of playing one from your hand, because there's no rider on the timing of its ability.
The funniest thing about this particular signature is that by the time you realize it doesn't say anything it's too late to stop reading it....
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