Yehuda Kelemer, former Rabbi of the Young Israel of Brookline, Massachusetts, shared the following account with Rabbi Aryeh Frimer and Rabbi Dov Frimer, who published it in Tradition (“Women’s Prayer Services – Theory and Practice I” 32:2 Winter 1998, p. 41). 

During the mid-1970s, one of the women congregants in the Young Israel of Brookline headed by R. Kelemer wanted to wear a tallit with tzitzit during the prayer services. After R. Kelemer had expressed to her his hesitations about the matter, she approached R. Joseph Soloveitchik, who lived in Brookline, to discuss the matter. 

Rabbi Soloveitchik explained that in light of the novelty of the action, it should be adopted gradually. He suggested that she first try wearing a tallit without tzitzit (which is, of course, permitted for women) and asked her to return to him after three months, at which time they would address the matter further. This approach seems to be in line with a responsum written by Rabbi Moshe Feinstein around the same time period.

When the two met once again, she described to R. Soloveitchik the elevated religious experience she had wearing the tallit. Rabbi Soloveitchik pointed out to the woman that wearing a tallit without tzitzit lacked any halachically authentic element of a mitzvah. It was obvious, therefore, that what generated her sense of “religious high” was not enhanced mitzvah fulfillment, but something else. Under such circumstances, he held, wearing a tallit was an inappropriate use of the mitzvah therefore he forbade the woman from wearing a tallit with tzitzit.

One reader who came across this story elsewhere online commented, "Sigh. Once again we are examining Women’s motivation and inspiration. In other words, do they deserve to be able to do something? They have to earn the right in some way."

In my opinion, although his remark may be valid from a social perspective, he failed to realize that there really is a clear halachic issue involved. The Shulchan Aruch weighs the question, and it all comes down to yehorah. Since this term is hard to translate into English, many people misinterpret the Shulchan Aruch on this point. The idea is that anytime one seeks to fulfill a mitzvah that he or she is not required to do, it becomes problematic based on the principal of yehorah, i.e. we want to avoid the appearance of doing a mitzvah in an ostentatious manner. This applies equally to men in other circumstances (e.g. wearing Rabbenu Tam tefillin). 

There is also a second fascinating account of a similar case. "I was a student at Maimonides during the 1970′s," wrote a woman by the name of Sarah Jacobs. "In 1974 I began wearing a tallit at my home shul. At the start of the school year in 1975 or 6 the school demanded that all students daven Shacharit at school. I informed members of the administration before the school year began that I would be wearing my tallit.

"I wore my tallit at t’fillot at Maimonides from that point until I graduated in June of 1978. I would assume that if the Rav [Rabbi Soloveitchik] had any objections to my wearing a tallit in his school I would have been told to stop wearing my tallit. In fact, the Rav used to daven at Maimonides when he was in Boston. I have no doubt that he both knew that I wore a tallit and had seen me wearing my tallit during t’fillot."

Personally, I don't see how his awareness of Ms. Jacob's practice is illuminating in any way, since she never asked him for his opinion. Most rabbis avoid challenging practices by individuals they may disagree with, except under extreme circumstances. 

Notably, Ms. Jacobs writes that several of her teachers had reported that Rabbi Soloveitchik's wife Tonya wore a tallit katan under her clothes (i.e. b'tzina).