doublets of doublets

I am still not able to resolve the doublet of doublets question that came up Tuesday. But since I promised, I am going to try to explain as well as I can what I'm thinking about it. If you are reading this, give yourself a break if you don't know how an nmr picks up signals. If you are reading this and do understand nmr, I would be delighted to have you help me improve my understanding.

Here's what I (think I) know:

Spin 1/2 nuclei, such as hydrogen, have spin that gives them magnetic moments.
The spin states in such nuclei are described as being either "up" or "down," and are quantized.
Inside the nmr, an applied magnetic field aligns spin states and then an applied radiofrequency signal can cause spin flips at particular frequencies. These spin flips cause absorbances which are converted to signals, generating the spectrum.
The chemical shift for a given set of equivalent hydrogens indicates what frequency is required to generate this absorbance, which varies depending on the environment that nucleus sits in.
Hydrogens that are nmr equivalent absorb at exactly the same frequency.

Still with me? Good. I am about to address splitting:

Absorbance peaks are split due to interactions that occur through bonds, but between nuclei of the hydrogen(s) generating a particular peak and those which are described as "nearest neighbors." We have learned how to identify nearest neighbors, so I won't go into that here.
The splitting occurs because the neighboring nuclei have a minor but real effect on chemical shift. Their spins can be in or out of phase with those that are generating the signal, adding to or subtracting from the chemical shift.
For every nucleus involved in these relationships, there is a particular number of possible spin-state alignments, which happens to equal the number of nearest neighbors plus one.
Therefore the peaks are split into that many (neighbor Hs plus 1) like we learned, and forms a set that varies in intensity like a Pascal's triangle.


Now, finally, for the confusion about the doublets of doublets:

Occasionally one hydrogen attached to a carbon is nmr inequivalent to another on the same carbon. For example, two hydrogens on an sp2-hybridized carbon could qualify, if the stuff on the other end of the alkene is distributed asymmetrically.

In this case it makes perfect sense to me that these hydrogens would each have a slightly different chemical shift, and that each one would be split by a nearest neighbor, generating an overlapping "doublet of doublets," rather than the triplet signal you would otherwise expect.

I can't understand the other example Straumanis gives, however, which involves hydrogens around an aromatic ring that has a single fluorine substituent. There are 3 groups of equivalent hydrogens on this structure: those at C1 adn C5, those at C2 and C4, and one at C3. He seems to argue that because the H at C1 exists in a different environment than the one at C4, that the signal from the H at C3 will split into a doublet of doublets.

If peak splitting is based purely on the spin-state interactions of the nuclei, what environment those nearest neighbor hydrogens are in doesn't seem to have anything at all to do with it.

Signals from hydrogens on aromatic rings do tend to get all overlappy and scrunched up, so he could be perfectly correct, but I can't sort out why. I think I need to send the question to him, or to a group of others I know are using the workbook.