I don't believe there is a physical limit to the number of values in a picklist. I recently tested how many values could be added to a Dynamic Application picklist field, and if memory serves, we got up to about 150 before stopping. However, I wouldn't recommend adding this many values to a picklist for a couple of reasons:
They must be manually entered and maintained, which is more of a hassle than an ongoing inconvenience but will be a source of frustration for someone, like when you lose all of your work if you hit the backspace key when the cursor focus is set outside of a text field, triggering the browser "Back" function.
In my experience, if users have more values to choose from than what is visible in the picklist without having to scroll, they will generally pick the value that is "close enough" or one of the first values in the list, or they won't use the form at all. From an adoption and usability standpoint, lists of values of more than about 20 options tend to be perceived as increasingly harder to use, or decreasingly intuitive.
Is there anyway to consolidate your list? Maybe group the departments at a level that makes it much easier as a picklist, then provide a text field for the specific department? This way you have the selectable data point for grouping and reporting, and the manually-entered level of detail the users can enter as appropriate.