A simple question with a very complicated answer.
As many people have no doubt mentioned above, you should always have a good idea of what to expect before relying on an analysis program.
I would probably divide jobs into simple categories as far as the overall members are concerned.
1. Simple spans and pinned connections - This I would always do by hand because it definitely tends to be quicker. I would even include double spans and more complex loading patterns in this analysis. The only exception for me would be when significant second order moments may be encountered.
2. Framed structures such as portal frames with significant secondary moments - I would tend to do an analysis on the computer and member checks by hand.
3. More complicated fixed structures with significant restraint, indeterminacy and possibly torsion. An example of this would be a complex canopy made in one welded 'lump'. I would either do a quick hand calc to determine sizes and then back it up with a computer analysis or I would do the analysis and refine it to the optimum solution then check the output against hand calcs.
Basically, you should always understand the simplifying assumptions of your hand calculations so that you can make the judgement whether you can live with it. Generally indeterminacy, secondary load paths, restraint, and second order effects are the main reasons why a hand calc would not be accurate.
As far as connections are concerned then it is usually the best approach to us the simplest method, it is easier for someone to check and is also easier to get it right.
Personally I have never used FEM and have only had one or two situations where I thought it would be truly essential.