How to cut a list by max sum

Split

ClearAll[split]
split[lst_, maxsum_] := Module[{s = lst[[1]]},
  Split[lst, Or[(s+= #2) <= maxsum, s = #2] &]]

Examples:

split[{5, 10, 3, 8, 9, 8, 8, 4, 1, 7}, 20]

 {{5, 10, 3}, {8, 9}, {8, 8, 4}, {1, 7}}

split[{1, 2, 3, 4, 6, 4, 3}, 20]

 {{1, 2, 3, 4, 6, 4}, {3}}

SequenceSplit

ClearAll[seqSplit]
seqSplit = SequenceSplit[#, a : {__} /; Total[a] <= #2 :> a] &;

seqSplit[{5, 10, 3, 8, 9, 8, 8, 4, 1, 7}, 20]  

{{5, 10, 3}, {8, 9}, {8, 8, 4}, {1, 7}}

seqSplit[{1, 2, 3, 4, 6, 4, 3}, 20] 

 {{1, 2, 3, 4, 6, 4}, {3}}

Reap + Sow

ClearAll[reapSow]
reapSow[lst_, maxsum_] := Module[{i = 0, s = 0}, 
  Last @ Reap[Scan[Sow[#, If[(s += #) <= maxsum, i, s = #; ++i]] &, lst]]]

reapSow[{5, 10, 3, 8, 9, 8, 8, 4, 1, 7}, 20]

{{5, 10, 3}, {8, 9}, {8, 8, 4}, {1, 7}}

reapSow[{1, 2, 3, 4, 6, 4, 3}, 20] 

 {{1, 2, 3, 4, 6, 4}, {3}}

ReplaceRepeated + TakeDrop + Accumulate

takeDrop[lst_, maxsum_] := {lst} //. {a___List, b_List} /; Length[b] > 1 :> 
      {a, ## & @@ TakeDrop[b, LengthWhile[Accumulate[b], # <= maxsum &]]} //
   DeleteCases[{}]

takeDrop[{5, 10, 3, 8, 9, 8, 8, 4, 1, 7}, 20]

 {{5, 10, 3}, {8, 9}, {8, 8, 4}, {1, 7}}

takeDrop[{1, 2, 3, 4, 6, 4, 3}, 20]

 {{1, 2, 3, 4, 6, 4}, {3}}