Testing for admissible sequences

Brachylog, 25 24 19 bytes

5 bytes thanks to Karl Napf.

lybb'(eM-yA,?:[M]z:%aodA)
l:2'(eM-yA,?:[M]z:%aodA)
l:2'(eMg:?rz:%adlM)

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l:2'(eMg:?rz:%adlM)
l:2                  Temp = [2:length(input)]
   '(             )  true if the following cannot be proven:
     eM                  M is an element of the interval
                         indicated by Temp, i.e. from 2
                         to the length of input inclusive,
       g:?rz:%adlM       every element of input modulo M
                         de-duplicated has length M.

Jelly, 10 bytes

JḊðḶḟ%@ð€Ạ

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               Input: L.
JḊ             Range of [2..len(L)].
  ð    ð€      For x in [2..len(L)]:
   Ḷ             [0..x-1] (residue classes)
    ḟ              without elements from
     %@            L % x.
         Ạ     All truthy (non-empty)?

Python, 61 60 bytes

q=lambda l,d=2:d>len(l)or q(l,d+1)&(len({v%d for v in l})<d)

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Edit: replaced logical and with bitwise & to save one byte