hi Stas. lots of questions

1. so if nothing is selected, are they supposed to have default values of 1-4, 5-8, 9-12?

2. i guess E3 is always 1-4? by the way, merged cells are always a bad idea. avoid it if possible

3. using your eg where 3 is selected for Choice Box 1, does Choice Box 2 contain 4-8 or 4-7? you mentioned 4 in each box only, but you said include 4, so i'm confirming.

4. you then say Choice Box 3 is 7-8? not 4 weeks? 7-10?

5. how about Choice Box 4? is it 8-12, disregarding the 4 per box rule?

for starters, you can select from E5:E10. using data validation list, try this formula:
=OFFSET($A$1,MATCH(E3,$A$2:$A$13,0)+1,,MIN(4,MAX($A:$A)-E3))

if choice box 4 is supposed to show 8-12, then use this for E9:
=OFFSET($A$1,MATCH(E7,$A$2:$A$13,0)+1,,MAX($A:$A)-E7)

by the way, your formulas in column F can be reduced greatly by using:
=VLOOKUP(E3,$A$2:$B$13,2,0)

or even without relying on column B:
=SUMIF($A$2:$A$13,"<="&E3)