To rigorously solve for deflections of indeterminate structures, the most feasible method is the Direct Stiffness Method (aka matrix methods). This is generally the approach finite element method software packages (FTool, SAP2000, RISA, etc.) are implementing when dealing with line elements.
For structures with a relatively small number of degrees of freedom it is possible to implement the direct stiffness method and arrive at a solution (somewhat) by hand. In my matrix methods class at university we generally solved the matrices using a calculator or by implementing a routine coded in MATLAB. So really, we were doing exactly what the FEM software does but more slowly and/or transparently.
Prior to the advent of computers (including programmable calculators) engineers would have used approximate methods of analysis to ballpark deflections. One very simple example of this might be that if we know the load on a beam we can determine an upper-bound deflection by considering it to be simply supported. When doing a hand check of computer calculations, engineers today will still take this approach. That is, we make simplifying assumptions and get a back-of-the-envelope feel for whether the computer solution is appropriate.
Computer-based analyses are fairly essential for the design of today's structures by today's design codes. The goal of hand calculations is rarely to exactly duplicate the computer solution. Rather, the goal is to use hand methods and engineering judgement to validate the computer solution.