Seeking Substantial Subcollections
Jelly, &nbsp;17 11&nbsp; 10 bytes

-1 Thanks to Dennis (use count to remove the need to tie atoms together with any quick)

Œ!ŒṖ€§Ẏċ€Ṁ


A dyadic link accepting a list and an integer which yields the maximal count, \$k\$.

Try it online! Or see the test-suite -- it's too slow for the larger test cases :(

How?

Œ!ŒṖ€§Ẏċ€Ṁ - Link: list L, integer I
Œ!         - all permutations
  ŒṖ€      - for €ach: all partitions (all ways to slice up each permutation)
     §     - sum each (vectorises at depth 1, so this gets the sums of the parts of
           -           each of the partitions for each permutation)
      Ẏ    - tighten  (to get a list of the sums of the parts of each possible partition)
       ċ€  - for €ach count occurrences of I
         Ṁ - maximum