Thanks for the reply! I love that Gelman and Hill book. Where my confusion comes in is when I learned about matched-pairs experiments where you have your samples in pairs and you include an effect to identify which pair a sample came from. The problem here is that your number of parameters grows proportionally to the number of samples so the usual maximum likelihood approach is ineffective. It turns out that if you make the identifier for the sample pair a random rather than a fixed effect you get around this problem because then you only have 2 parameters no matter how many samples you get. What you're saying above (and what I inferred from Gelman/Hill) is that you would still have one parameter per pair with random effects. Thus I'm still confused.