The first agent who offered representation on my manuscript had the partial for a few weeks before requesting the full. I let her know that two other agents were considering the full, and she promised to read it as fast as she could. She actually called me 12-13 hours later (without having quite finished the book) to offer representation. VERY exciting and flattering to say the least! I then let the two other agents looking at fulls know, as well as the agent looking at the partial who was one of my very top choice agents. The other two agents with fulls rushed me to the top of their piles, although both ended up passing (one said she would have to ask for very extensive revisions and even then couldn't promise an offer--her revisions would have made it a completely different book which helped make my decision easier). The agent with the partial promptly congratulated me on the offer and asked for the full manuscript, along with three days to read the manuscript, which I was more than willing to grant.
Of course, as angst-filled as many of us writers tend to be, I had convinced myself that no way was she going to offer, too; but then I was lucky enough that she did! Also without quite finishing the manuscript, but she wanted to be sure to respond to me when she promised, and loved the manuscript so wanted to make the offer. After a couple of days of more angsting, and feeling guilty that I had to disappoint one of the two agents, I signed with the second for several reasons. I was honestly sad to say no to the first, though!
Which, I suppose, is a really long way for me to tell my story!