Yes, they can. The board has discretion over the allocation of the options pool that will have been set aside as part of each round.
However to issue those to yourself would be like eating your seed stock, since that's the pool that you use for issuing options to new hires, and without that you can't give new employees any equity. For that reason I don't think it's likely.
I think what the grandparent comment is getting at is that you as an employee have little control over the delayed compensation strategy. If you're lucky, you have a honest founder and investors who make sure you're paid for your contribution at deal closing time. If you're not lucky, you have a board/CEO/founder that will take whatever they can get away with (e.g. your value add) and then point to the financial rules/contingencies and say, "Well, we tried to do all we could, but we had to do this to ensure the success of the company. We needed to compensate the administration because it's hard to find such good talent like ourselves. Your still getting something here..." Or some such line. And you end up with some minuscule share at the same time providing critical value to the business.
However to issue those to yourself would be like eating your seed stock, since that's the pool that you use for issuing options to new hires, and without that you can't give new employees any equity. For that reason I don't think it's likely.