Learn English – “Instructions on” vs. “instructions for”

If I absolutely had to choose between these two alternatives, I would pick "limited between", but generally speaking, I would go with "limited to", as in:

The number of weeks for data retrieval is limited to 25–52.

(Notes: that's an en dash, I've added a the, and I don't repeat the word weeks.)

The British National Corpus does not have a single cite for "limited between" or "limited within", but it has 1343 cites for "limited to", including several cites for value ranges, e.g.:

• [...] limited to 1 to 1.5p on a pint of beer [...]
• [...] limited to 15 to 18 per cent for a transitional period [...]

It even has two cites for "limited to between" (but again, none for just "limited between").

In conversational use, you will encounter both expressions and, as brick notes, provided that which is slightly more formal.

In a scientific paper, I would avoid all of these forms. The link subject to the requirement that makes the status of the caveat crystal clear. If you wanted a less verbose expression, you might try so long as or even if or when.