It depends on what you know about the board. You can do a lot more if you have schematics and layout drawings and chip datasheets than you can without.
The vintage and integration level of the parts makes a big difference. If you have a microprocessor with only 8 pins and the only thing coming off the chip is I/O then there isn't much to look at. If it's an older board with external address decode logic, timing one-shots, and other glue logic then you can check all that stuff if you have the necessary information. Old time depot repair guys had tons of tricks for quickly checking for common failures. One trick that really impressed me in the early 80's was a microprocessor chip whose data bus had been strapped to a NOP (no operation) instruction so that it would just count through all of the address range for checking address decode logic, etc. I don't know if doing that to a CPU for one board would have been cost effective, but since they were using the same microprocessor chip on everything it worked out great.
As far as testing 74xx glue logic goes you can easily see if the outputs are going up and down with a scope, and see if they match the inputs (at least for the simple chips). A four channel scope lets you check gates and small decoders/multiplexers but you need more channels to really test bigger chips (think logic analyzer). In practice, the typical depot repair guy in the 80's had access to a 2 channel scope and it was cheaper to replace a chip that acted funny than it was to spend tons of time trying to figure out what was wrong with it. Most of the places I worked had somebody whose job it was to write manufacturing test software that tested individual circuits and that software was also often useful for depot repair diagnosis.
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