In most cases of throat infections, your tonsils get swollen up. At this point, most doctors suggest that you get them removed as they cause unnecessary pain and discomfort. They consider it another "useless" organ like the Appendix, and thus want to get it out of the way.
Rather than being a cause for concern, the tonsils actually act as a barrier. They prevent whatever infection you have incurred from entering your digestive system, thus keeping any serious problems at bay. But they get swollen up in return.
Now, you must have noticed that in India, it is considered bad manners to sip the glass/tumbler. Such a glass would be called "Jhutha", and become invalid for further use unless it is properly washed. The basic reason that will come to your mind is that why should someone drink from a used glass? But there is a deeper significance...
Though, the primary reason is to avoid any infections to be communicated from one person to the other, another factor that must be considered is the regular cleansing of your tonsils. When you drink water bending your neck and let it flow from top, it obviously flows over your tonsils and cleans them. Thus, it will wash away unneccasary food particles and maintain your throat in order.