Can You Leverage An Offer Too Early?

Hi all, first post.

Graduating this year. I have been told that you can sometimes get preferential treatment if you let prospective employers know that you have an offer elsewhere. I have a final-round assessment centre in a few weeks for a good role at a good company but a foreign office. I am getting ahead of myself with this post but if, fingers crossed, things go well and I land this this job, would it be too early in the recruitment cycle to go to other companies I've applied to and try to leverage them to give an offer quickly? A lot of other companies will not have even closed their CV intake by the time I hear back from this place, so I feel like I would be more likely to hear a 'good for you' from HR, and taken out of the running. Let me know if anyone has experience with this. Thanks in advance everyone.