To spin off from NReilingh's answer, the half-tone between G and F# requires less extra tubing to be in tune, than the half-tone between E and Eb.
I believe that the third valve's length often is determined to have Eb in tune, and is therefore a little too long the have an in-tune E.
Let the tubing required to get from G to Gb be a reference, let's say value 1.
Gb -- 1 -- G
The subsequent half-tones downward grow larger the lower you go. Say for example that the required extra tubing to get from Gb to F is of length 1.01 instead of 1 (probably a wrong value, but the principle remains).
F -- 1.01 -- Gb -- 1 -- G
And the lower the half-tones, the more extra tubing is required.
Db -- 1.05 -- D -- 1.04 -- Eb -- 1.03 -- E -- 1.02 -- F -- 1.01 -- Gb -- 1 -- G
Perfect tuning in order to go from:
- G to F is 1 + 1.01 = 2.01
- G to E is 1 + 1.01 + 1.02 = 3.03
- G to Eb is 1 + 1.01 + 1.02 + 1.03 = 4.06
Say that your first and second valves were made in tune. Valve 2 (G to Gb) would have a length of 1. Valve 1 (G to F) would have a length of 2.01.
Now to get from G to E, you can combine 1 and 2, and get tubing for 1 + 2.01 = 3.01. It is a little short since in-tune tubing requires 3.03.
To get from G to Eb, you can combine 2 and 3. The third valve length is not determined yet, you can build it so that 2+3 has in-tune length of 4.06. The 3rd valve has then length 4.06 - 1 = 3.06.
So here you are, with valves set for in-tune Gb, F, and Eb, the tubing length for E is:
- in-tune -> 3.03
- 1+2 -> 3.01
- 3 -> 3.06
1+2 is closer to being in-tune. It is also at a higher pitch, and it is much easier to compensate with your lips downwards than upwards.
In real cases, I suppose that none of the valves really are in tune, there are many compromises.
All this letting alone the dexterity issue: pressing the third valve alone is often more difficult than 1+2. This is not always the case of course.