Yes, it is possible to build a cell phone that can perform the functions of making calls, sending SMS, and even data transfer. For an example of a GSM module that would be the main component of such a system, see here: http://www.sparkfun.com/products/10138
However, from a cost and performance point of view, it doesn't make any sense to do so. The cell phone OEMs have a large amount of engineering and logistics resources to put behind building devices that are aesthetically pleasing, small, and powerful.
You can and a simple Google search will link you to a number of homebrew projects.
Hint: include terms like "avr", "Arduino", or "arm" in your searches.
The problem with home brew cellular systems is, unless you have access to some pretty advanced chip/board fab equipment, they're usually way too bulky to be practical, and they use too many mAh's (battery life).
The stuff on Sparkfun and Adafruit are great for hobbyist projects, but aren't going to be better than a cheap series 40 Nokia that's hackable via a serial interface.
major chip manufacturers probably have reference designs you can get, which include some of their parts, and which should be a good starting point.
you will indeed need to have access to advanced PCB fab and assembly subcontractor, but with some digging around you can find those.
the biggest problem is that you'll need hardware and software engineers who really know their job and can do the schematics, layout and firmware - doing a cell phone well is not an easy or small project, certainly not if you want to do something at a serious level and go beyond hobbyist type hacking.
you'll also have to do research on where to get an OS - developing this yourself is a huge project, probably just as big if not bigger than doing the hardware design. I don't know if it's possible, but perhaps you can get android OS source code and use that as a starting point.
you'll have to make each part of the hardware talk to the OS which means writing some driver or glue code, which is a whole other job. I'd try to use standard peripherals/ICs in your design for which android already has drivers.
without knowing more about your project, I can't really help more than this - if you want to talk about it feel free to msg me.
However, from a cost and performance point of view, it doesn't make any sense to do so. The cell phone OEMs have a large amount of engineering and logistics resources to put behind building devices that are aesthetically pleasing, small, and powerful.