The answer is yes, but with a possible downside. Whitening fringe is a fairly inexpensive process but could have long term consequences to the value of your rug. Often after washing a carpet, the fringe is bleached to accent the overall wash and by producing a fresh look to the fringe itself.
Although cotton fringe has been used in carpets for many years, nothing has changed about the fiber itself. As organic matter, cotton fibers degrade under the use of harsh chemicals such as bleach. While bleach won't necessarily disintegrate your fringe, the use is not recommended as final results are fairly inconsistent. If fringe is over-bleached, or the bleach had not been neutralized/properly washed out, it will cause fringe deterioration. Remember, on an oriental rug, the fringe is not just decoration, it is a continuation of the rug's structure. Should excess bleach be absorbed past the visible fringe, the structural integrity of your rug may be compromised. Unnecessary loss of fringe as described can decrease the value of your rug. If the fringe is worn, the rug looses its buffer, and knots towards the ends can easily be lost.
Giving Your Fringe the tug test
In the picture below, I'm testing the fringe of a carpet recently returned from washing. To test the fringe yourself, take one single tassel in an inconspicuous area. Don't yank hard, simply apply slow and steady pressure. If your fringe is in good shape, it will not release even if strong force is applied. Conversely, if the tassel releases with little to slight effort, the fringe is not in good shape. This single tassel popped off a split second after the photograph was taken.
Overall, this fringe is healthy. There were areas of the fringe which reacted poorly to the light "tug" test. It didn't take much for this single tassel to pop off of the rug. A fairly significant area around where this sample was taken had been badly degraded by the over-use of bleach.