If the alternator and battery were really tested thoroughly, then the problem is most likely either un-noticed corrosion on the battery terminals; or poor ground connections between the battery and the engine block; or a resistive or missing ground cable between the engine block and the firewall; or poor contact between the positive battery cable and the power distribution point at the starter solenoid. If the ignition switch internal contacts have become oxidized or worn; this can also create enough voltage drop to cause such problems. For that reason, it may be valuable to measure the voltage drop with the engine running, between the positive battery terminal and those fuses which are only hot when the key is on. The supply voltage at the ignition system should also be checked with the engine running and compared with the voltage across the battery. When the engine is running, there should be less than 1/2 volt drop between the battery and any loads.
Many people will test a battery with a voltmeter; which really does not provide enough information to evaluate the battery's overall condition. A battery load test is the only reliable way to check a battery. Similarly, most tests of alternators will not check for bad diodes. Specialized equipment and knowledge is required to test alternators for bad diodes.