In Harry Potter, can a wand run out of power, i.e. cast all the magic it can possibly perform and then just break, or go dead?
The magic comes from the wizard, not the wand. As Ollivander says:
Oh yes, if you are any wizard at all you will be able to channel your magic through almost any instrument.
Harry Potter and the Deathly Hallows - Chapter 24, The Wandmaker - Page 402 - Bloomsbury
So unless the wizard runs out of power, which I very highly doubt, no.
A wand can hold residual magic, that will drain, but that doesn't really count...
Muggles cannot perform magic. The best - or worst - they could hope for are random and uncontrollable effects generated by a genuine magical wand, which, as an instrument through which magic is supposed to be channelled, sometimes holds residual power that it may discharge at odd moments
The Tales of Beedle the Bard, page 82
It is possible.
I agree with @Mithrandir's answer on the magic mainly coming from the wizard, however there is this point from Pottermore.com's section on wands. (Emphasis mine):
Minor disadvantages of unicorn hair are that they do not make the most powerful wands (although the wand wood may compensate) and that they are prone to melancholy if seriously mishandled, meaning that the hair may 'die' and need replacing.
The combination of wand wood and the magical substance at the core of a wand produce a delicate magical instrument which channels the wizard's power in a specific way. If a wand with a unicorn hair core is damaged or mishandled and the core 'dies' and indeed needs replacing, it would be accurate to say that this particular wand had run out of power and gone dead. It is also possible for a wand to 'die' at the death of their owner in the case of the Hazel wand:
It is so devoted to its owner that it often ‘wilts’ (which is to say, it expels all its magic and refuses to perform, often necessitating the extraction of the core and its insertion into another casing, if the wand is still required) at the end of its master’s life (if the core is unicorn hair, however, there is no hope; the wand will almost certainly have ‘died’).