list = {5, 10, 3, 8, 9, 8, 8, 4, 1, 7}; 

Using SequenceReplace:

SequenceReplace[list, {a__} /; Total[{a}] <= 20 :> {a}] 

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

The sister version of SequenceCases, which is also not fast for large lists, but is short and clear.

list = {5, 10, 3, 8, 9, 8, 8, 4, 1, 7}; 

A variant of rhermans SequenceSplit using SequenceReplace:

SequenceReplace[list, {a__} /; Total[{a}] <= 20 :> {a}] 

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

The sister version of SequenceCases, which is also not fast for large lists, but is short and clear.