[nimk]

So I made a buffer solution (using a salt salt, pKa 7.21) and added a concentrated strong base to it to achieve a certain pH (7.27). I then added some deionized water to get it to a final volume of 100 ml. As I did so, the pH increased to 8.12. Why did it change? From what I've read, pH should not be affected by water...

[Borek]

You must have made some mistake.

pH of buffers is not completely unaffected by dilution. Changes are minute, but they are there. After all, once you dilute the buffer infinitely you are left with just pure water, so naively you could expect on dilution the pH should drift to 7.00. But it should never go past 7.00 like you describe.

Actually DI water is never perfectly neutral, even just opening the bottle is enough for atmospheric CO₂ to dissolve and change pH to slightly acidic. That makes predictions of changes in pH on dilution pretty hard, but 8.21 is definitely not something to be expected.

[Babcock_Hall]

Just to amplify Borek's answer, the apparent values of pK_a are affected by something called the ionic strength.  Ionic strength is related to the concentration of charged species in solution.