Toyota Repair: starting problem, toyota camry v6, toyota camry

Question
I have an '89 Toyota camry V6 which has 155,000 mi and has a starting problem. I just had it repaired at a toyota dealership which replaced the ignition switch and previously had fixed the wire harness telling me that the problem might occur again which I have since had them recheck again to see if there is another problem. Over the summer the car was starting okay, but as soon as the weather has started getting colder, the problem has reappeared with it not starting intermittently. I have to turn the key a number of times and it would start until after maybe 5-10 turns of the key. Yesterday the rainy and cold weather contributed to it not starting and I had to jump start it. After running it for 15 mins, I shut it off and it was not starting and I had to jump start it again and run it for 45 mins to warm it up. The battery is relatively new and the alternator was changed  as well as the starter. If I take it to be checked out, they say that the battery and alternator checks out okay. The mechanic says tht there is a loose connection somewhere, but nobody is able to figure out what the problem is. Can you comment on the what I need to check out in order to fix the problem.  

Answer
From what your saying is it cranks so slow you have to jump it off, but the battery is new,starter& alt, I would ceck the cables coming off of the battery as I have seen these corrode way down in the insulation, making you think there good but only having a bunch of powder and very few wires left and I would think they checked the relays, check the ground to eng/trany( not just look but remove to see if you have corrosion under the eyelets, This is a good test, turn your lights on and try to get it to do this, if they dim, like a dead or weak battery, check to make sure its not dead or weak then do the above.