Say we have two inertial observers Bob and Alice moving relative to each other at a significant fraction of the speed of light c (say 0.5c). Bob is moving to the left relative to Alice and Alice to the right relative to Bob.

When their origins cross, a pulse of light is emitted to the right. In Bob's frame, he sees Alice moving in the same direction as the light pulse at 0.5c so thinks she must see the light moving away from her at 0.5c. She says she sees light moving at c, and so Bob accuses her of using shorter metre rulers in measuring the distance travelled by light in her frame (I'm considering length contraction and time dilation independently for the moment).
So if she measures some length in her frame that looks the same as what Bob is measuring, she gets a larger numerical value than what Bob is getting.

Now in Alice's frame, she sees Bob moving away from the light pulse at 0.5c so she concludes Bob must see the light moving away from him at 1.5c but he insists the he sees the light pulse moving away at c. So if she accuses him of using shorter rulers to measure the distance travelled by light in his frame. But if he's using shorter metre rulers than shouldn't he measure a larger numerical value for the distance and calculate a larger speed of of light than what Alice gets?
(i.e. he measures the same distance as Alice measures but because his rulers are shorter, he gets a larger number for distance).

What've I misunderstood here? (this is just some preamble before going into the derivation of the Lorentz factor by the way*. An example like the above was given but only for someone going in the same direction as the speed of light as in the second paragraph and I wanted to make sure that it would work the other way around)

You are considering length contraction but leaving out the effects of time dilation and relativity of simultaneity - they all play together to produce internally consistent results.

To see what's going on more clearly, you can try one or both of:
1) Drawing a space-time diagram (many examples in previous threads - look especially for posts by ghwellsjr).
2) Write down the x and t coordinates of the light-emitted/light-received events, using the frames of either observer. It's most convenient to start with Bob coordinates for flash-reaches-Bob event, and Alice cooordinates for the flash-reaches-Alice event. Than use the Lorentz transformations to convert to the other frame, see when and where the other observers claims these events happened.

Ah ok, thanks. It was just because in the lecture he said 'or' when referring to length contraction and time dilation so I took it to mean there one or the other could happen independently, or both at the same time. I'll try the suggestions.

The confusion comes about because it's sometimes possible to consider only one but still come up with a pretty decent (but incomplete) analysis of a physical system from the standpoint of a particular observer.

For example, you'll see explanations of the muon delay measurements that say that the decay of a fast-moving muon is slowed enough by time dilation that the muon will live long enough to make it through the earth's atmosphere and reach the ground before it decays. That's true, but it's incomplete because it only works for an observer at rest on the surface of the earth (which is where the experimenter's lab is, which is why you hear this explanation so often).

From the muon's point of view, it's living a normal undilated lifetime, and it is length contraction of the distance between the top of the earth's atmosphere and the ground that allows it to live long enough to be hit by the ground rushing towards it. That's also true, but incomplete because the analysis only works for the muon.

Neither analysis will satisfy an observer moving relative to both the muon and the earth - he'll have to consider both time dilation and length contraction, and all three will also need to consider the relativity of simultaneity if they're going to compare their notes and see that they're all observing the same physical situation and getting results that are consistent with each other and what we know of muon behavior.