Not just any hat, but a specific one.
I met a friend for dinner last night, and since he works later than I do, I wandered around for a bit until it was time to take the subway (this was pre-Typhoon last night). During my meanderings I found a really cute hat. An inexpensive, really cute hat. I tried it on, and it looked terrific.
So, what's the problem? Hats are a loaded symbol in observant Judaism. Many women cover their hair after marriage, and choose to do so with hats or scarves or wigs (I will NOT be wearing a wig, I agree with Rabbi Riskin on this one). As a result of this, many young woman avoid wearing hats, because it might lead people to mistakenly assume that they are married (and what if you meet your soulmate, but you're wearing a hat, and he thinks that you're taken? You totally missed your chance. That was it, now you'll die alone.) Same thing with broad headbands (often used to secure wigs).
Also, it sends the statement "I am preparing for marriage." As in "I will get married soon, so I better stock up on hats."
So, to recap, I can't buy a cute hat because a) it might make people think that I'm married, or b) it'll freak out my boyfriend*.
* One of my soon-to-be-married friends offered to let me store hats at her house until the time comes.