Like any good morality tale, there was a flaw in life's march to perfection. The early nervous system could integrate patterns distributed across multiple sensory dimensions, but it couldn't integrate patterns distributed across time. Your basic Triassic dinosaur could easily recognize the ominous sound of heavy footfalls, but he couldn't figure out that increasing amplitude of footfalls meant approaching danger, and decreasing amplitude of footfalls meant receding danger. Obviously, such an ability would be most conducive to survival, but the basic layout of nervous systems didn't handle this problem very well.
Let's examine closely the problem of recognizing the difference between approaching footfalls and receding footfalls. Let's say that one requires at least four footfalls in sequence, with the amplitude of each footfall in the sequence to be higher than the amplitude of the previous footfall. In our stupid little diagramming system, the neural circuit for this function might look like this:
The bottom neuron won't fire unless all four inputs trigger. But notice that, should they trigger at the instant the sound is heard, they will fire in sequence, and the bottom neuron will never get all four inputs together. It will get one input and then a long, long time later (in neuron years), it will get the next input, and so on. The only way to solve this problem is for the input neurons to remember their inputs for some period of time, and then all fire together. And just how can we enable neurons to remember? This may seem like a trivial problem -- after all, since we humans enjoy gobs and gobs of memory, and even the stupidest computer comes equipped with megabytes of memory, there really shouldn't be much of a problem getting memory out of neurons, right?
Wrong. Yes, it is possible to set up long-term memories by permanently etching patterns into the connections between neurons, the weights of the various inputs, or the trigger thresholds of each neuron. But that requires significant anatomical changes inside the brain. If one footfall comes just one second after the previous footfall, there just isn't enough time to grow the extra dendrites or change the neuronal chemistry to remember the previous footfall.
What we need here is short-term memory, and that particular kind of memory can only be set up with neurons using circulating neuron sequences. Neuron A triggers Neuron B which triggers Neuron C which triggers Neuron A and the cycle starts all over again. But it's much more complicated than that. You need additional neurons to specify exactly what that little circuit means, what part of the brain it came from, and what part of the brain it should go to, and when it's time to forget that tidbit of information. Whatever the scheme looks like, it's unquestionably complicated, and requires plenty of neurons to remember each tidbit of information.
An analogy from electronics
Interestingly enough, there's a parallel problem in electronics. If you want to built a simple pattern recognizing circuit, it takes just a handful of simple circuits called gates. A circuit to recognize an eight-bit pattern might look like this:

This little circuit will trigger when it gets the pattern 10110110, which is equal to the number 182, as an input. However, should you wish to transmit this number using a standard serial protocol such as RS-232, then your circuitry suddenly becomes vastly more complicated:

Mind you, this is not an actual circuit diagram like the one above; this is a block diagram showing only blocks, each of which represents large circuits containing many gates, shift registers, and other components. The point is that serial processing demands much more complexity than parallel processing.
Serial processing in animals
Sometime during the Triassic, animals started developing the capability for serial processing. It's impossible to know much about the history, but we know that mammals, which had their origins in the Triassic, all possess some capability for serial processing, whereas reptiles and amphibians don't. Birds, which got their start somewhat later, show mixed abilities for serial processing. In general, they're pretty bad at it, but in certain specifics their serial processing capabilities are excellent. The dinosaurs themselves never bothered to pick up on the advantages of serial processing. When you're even bigger than a 500-pound gorilla, you don't need no stinkin' geek-mammal tricks.
Here's a simple, utterly unscientific test for serial processing capabilities. Place a test animal inside a pen. Then open a gap in the pen, making certain that the animal is aware of the gap, but don't allow the animal to escape just yet. Entice the animal with a delectable treat to the opposite side of the pen, where you prominently place the treat on the outside of the pen. Question: how long will it take the animal to figure out how to get the treat? The problem requires serial reasoning; the animal must figure out the path -- the sequence of steps -- necessary to reach the treat.
If it's one of the smarter mammals, such as a dog, cat, or pig, it will figure out the solution almost immediately. An herbivore, such as a goat, cow, or horse, will take a little longer. A duck, chicken, turkey, or emu will never figure it out.
Another example of serial reasoning is the evasion path taken by prey attempting to escape from a predator. Most mammals will zig and zag as they run, trying to throw the predator off balance, and they will take terrain factors into consideration as they run. In practical terms, this means that ducks and chickens are easy to herd, but goats or pigs require more cunning and some fast moves.
Of course, those bigger brains didn't come cheaply; on the contrary, brain matter is the most expensive kind of tissue an animal can have. One pound of brains gobbles up more nutrition than a pound of muscle, kidney, or skin. So animals with bigger brains must eat more food. The Darwinian question is, do the performance improvements offered by bigger brains justify all that extra food that must be eaten? Eventually, the mammals figured out how to answer that question in the affirmative.
Our next exciting episode: Here Come the Primates!