The combined weight lift

1. with 1 or 4 boxes: it would imply a box of 100 or 189, again incompatible with the observed pairwise sums.
2. We discard 100: with 2 boxes: a pair of 100 would be needed (it does not exist).
3. We discard 100: with 3 boxes: the complementary pair should add up to 289-100=189 (impossible).

Back to problem
Answer: No. There is no selection that adds up to exactly 100 kg or 150 kg.
Let $w_1,\dots,w_5$ be the weight of the five boxes.
The sum of the 10 pairs noted is:

$$ 110+112+113+114+115+116+117+118+120+121=1156. $$

In the sum of all pairs, each box appears exactly 4 times, then:

$$ 4(w_1+\cdots+w_5)=1156 \Rightarrow w_1+\cdots+w_5=289. $$

Now we discard 150:

• with 2 boxes: a pair that adds up to 150 would be needed (does not exist),
• with 3 boxes: the complementary pair should add up to $289-150=139$ (does not exist),
• with 1 or 4 boxes: it would imply a box of 150 or 139, incompatible with all observed pairwise sums being between 110 and 121.

We discard 100:

• with 2 boxes: a pair of 100 would be needed (does not exist),
• with 3 boxes: the complementary pair should add up to $289-100=189$ (impossible),
• with 1 or 4 boxes: it would imply a box of 100 or 189, again incompatible with the observed pairwise sums.

Conclusion: The forklift cannot start with any combination of those boxes.