Does Quantum Mechanics say that anything is possible? I may be incorrect in this, but doesn't Quantum Mechanics state that everything has a probability of occurring (with some probabilities so low that it will essentially never happen)?
And if I am correct in my thinking, could it mean that, quite literally, anything has a possibility despite a potentially low probability?
(A version of my idea, which I admit closely borders on fictional magic, would be:
It is completely possible to walk through a wall, but the probability is so low that you need to walk into a wall for eternity before it can be done.)
 A: Quantum Mechanics does not say that everything is possible. In fact, it says that certain things are impossible. For example, a bound electron orbiting a hydrogen atom can only be measured to have certain discrete energy values with no possibility of measuring things in-between.
Although your example is right, quantum mechanics does tell us that if you walk into a wall for long enough eventually you might quantum tunnel through it, but that the probability is so that in practicality its not even really worth considering. Even the probability of a single atom tunneling through a solid wall is going to be extremely low.
A: Yes, of course so long as it obeys the rules of quantum mechanics. That is, you can't say that "everything is possible, so the possibility that quantum mechanics is wrong might occur, so it is wrong". Those are not the kind of predictions quantum mechanics can make. 
Let's look at the kind of predictions quantum mechanics can make. To do that, let's consider the quantum information stored in your brain. If all the important information in your brain counts on the positions of $10^{23}$ "particles", then the thing encoding all the knowable quantum information about your brain is a function of $3\times 2\times 10^{23}$ variables. That's a huge thing! It describes all the quantum weirdness of your head interacting and being entangled with the rest of the universe.
That thing is the "density matrix". A density matrix is like a quantum wavefunction, but I use it here because your head is entangled with the rest of the universe and I don't want to include the rest of the universe in my wavefunction.
You want to know what the probability is to measure your brain in a state where, say, your neurons are in a state such that you have lifelike memories of being raised by a wolf and then abducted by aliens. Well let's say your head fits in $1 m^3$ of space. Then the space of all possible configurations has a volume: $$1 m^{6\times 10^{23}}$$
(Not meters cubed/meters to the third power! Meters to the six times ten to the twenty three)
First, to be clear about something, let me go through an example with a dartboard first. What is the probability of hitting the spot $0.5$ meters from the left of the dartboard if you throw a dart that lands somewhere between $0$ and $1$ meters from the left of the dartboard? Well, the probability to hit that exact point is zero. If your dart throws happen to be uniformly distributed, then the probability of throwing a dart within an interval of length $x$, is $x$. You need some parameter $x$ depending on how specific you want to be. I'm going to do the same thing below, with an $x$ that depends on how specific you want to be. 
Denote that big number $6\times 10^{23}$ by $y$. 
If you want to know the probability of you having that memory, plus or minus a few memories (that's where $x$ comes in), you need to divide the volume in phase space of having that memory: $x^{y}\mathrm{m}^y$, by the total volume in phase space the brain can occupy, $1 \mathrm{m}^y$. This tells you that the probability of having a state with those memories is somewhere around
$$x^{10^{24}}$$
where $x$ is any number you like. It could be $.9$ for all I care. You still get a probability on the order of:
$$10^{-10^{24}}$$
For reference, there are about $10^{8\times 10}\approx 10^{10^2}$ atoms in the observable universe.
A: You have probably heard a garbled version of Murray Gell-Mann's totalitarian principle:

Everything not forbidden is compulsory

In quantum mechanics some processes are forbidden usually because they violate conservation laws. This is what CuriousOne refers to in his comment. Gell-Mann's principle states that unless a process is expressly forbidden there is a non-zero probability that it will occur. That probability may be exceedingly small, but in principle if we wait long enough the process will occur.
As CuriousOne says, this does not mean anything is possible because many things are expressly forbidden.
Re walking through a wall: you and the wall are too large to be easily described using quantum mechanics. At this scale decoherence is effectively instantaneous and you need to take this into account when calculating the behaviour of the system. You cannot apply a simple tunneling calculation to get the probability that you will tunnel through the wall. having said this, I would guess the probability that you will pass through the wall is greater than zero, though by such a small amount that you would need to try repeatedly for many times the age of the universe for it to happen.
A: The short and direct answer to your main question is no, quantum mechanics does not say that anything is possible.
In regards to the "walking through a wall" question, the answer is not as you say, "walk for eternity." All you have to do is live long enough for the wall to crumble, then you walk through.  This would be much shorter than eternity!  
