Are linearly dependent contrasts permitted when performing planned comparisons in within-subjects ANOVA? 
*

*Here it says that planned comparisons should be linearly independent. 

*Here (page 13) and elsewhere I see people giving examples of linearly dependent contrasts. 


How should this conflicting advice be reconciled? 
 A: Except in a designed experiment in which everything is perfectly balanced (and everything works perfectly, so that it stays balanced in the end), the set of contrasts of interest are not likely to be linearly independent.  There's nothing wrong with having linearly dependent contrasts, and I would resist the idea that planned comparisons must be linearly independent.
At the page you linked to it said:

Good researchers will list all of the planned comparisons that seem
  theoretically reasonable and then check to see which ones are
  independent of one another. Armed with that information, they select
  the planned comparisons that are most theoretically important and also
  independent of all the other planned comparisons they intend to run.

I think that's a bit silly.  The planned comparisons should be the ones that are interesting to you, linearly independent or not.
The whole "planned comparisons" business has to do with accounting for multiple testing.  And so the only reason that I can think of for requiring planned comparisons to be linearly independent is to simplify the adjustment for multiple testing, but I don't think that's a legitimate reason...just a convenient one.
A: Orthogonal contrasts have non-overlapping, complementary effects (in balanced design): whether the groups differ significantly or not in one contrast doesn't impact whether the groups differ significantly or not in another contrast, orthogonal to it. This is convenient to interpret.
Contrasts need not be orthogonal and often they are not. Actually, "contrasts" is a set of "contrast variables" to which a categorical predictor is being recoded (internally) by statistical procedure (such as GLM). Famous dummy variables are an example of contast variables, and they don't make orthogonal contrasts: if you recode your categorical predictor consisting of equally sized k groups into k-1 set of dummy variables, they will correlate with each other. But if you recode to Helmert variables (they constitute orthogonal contrasts) they won't correlate.
A: If your planned comparisons are linearly independent (orthogonal), you can run multiple contrasts without adjusting your alpha level. If your contrasts are not orthogonal, you have to correct for multiple comparisons.
You can use Helmert's procedure to generate a list of orthogonal contrasts.
Re: Jeromy's comment, I found this tidbit online @ this website:

For a priori planned orthogonal contrasts, the conceptual unit for error rate is conventionally taken to be the individual contrast (rather than the family of contrasts in the full set), just as it is taken to be the individual term in multi-factorial ANOVA partitioned into treatment effects and interactions (rather than the full experiment). The family-wise Type-I error must apply, however, if contrasts are used for post hoc comparisons to locate the biggest differences amongst levels of a treatment. The family-wise error rate for m independent tests, each with an individual error rate α, is 1 - (1 - α)m; the family-wise error rate for m orthogonal contrasts is some small amount less than this because their significance tests are not independent (since all use the same error mean square, even though the contrasts are independent since orthogonal). The size of α can be reduced to control the family-wise error rate, though at a cost of substantially diminishing power to detect individual differences.

In any case, I'm think this is what the source is referring to when it says your contrasts should be orthogonal. Even if, as Karl pointed out, the source is being conservative at the expense of pragmatism.
