Proper Component Testing (Specifically Resistors)
If resistors in your product are failing in the field, there are a few things I'd look at:


Are you buying parts from a reputable vendor? If you're buying from E-bay or a 2nd-tier distributor, you don't know how the parts have been stored and handled, and what quality controls were used when manufacturing them, or even if they are correctly labelled.
Is your design using the parts within their specs? If you're buying from a reputable vendor, an overstress due to a design error is the most likely cause of field failures. Is the operating temperature (including self-heating) within spec? Is the operating current in spec at all times? Are you de-rating the parts appropriately for the lifetime you need?
Is your manufacturing facility handling the parts correctly and following recommended thermal profiles for these parts? Are they following requirements for moisture sensitivity? 
Are your customers using your system within specs in the field? Are they using it under the temperature and vibration conditions you expected? Are they using the product at higher altitude than you expected?


If you don't find any issues by answering these questions, then is the time to start looking at testing. (of course, if you've had customer-visible failures, your customers may insist on further testing after you take corrective actions for whatever issues you found through design analysis)

For a problem like this, HALT/HAST testing ("highly accelerated life testing" and "highly accelerated stress testing") could be appropriate. This involves operating your system in extremely aggressive environments, with high temperature, humidity, and ongoing vibration. The environmental stress is increased until the product fails. The failure mode is analyzed and corrected, and then stress is increased until the next failure mode is observed, and so on. The idea is to find the weakest areas of the design and improve them, iteratively, until some fundamental limitation is found or increased stress becomes impractical.

But HALT/HAST is a system-level test, and you asked for a component-level test. For component level, accelerated life testing is also possible. Generally, based on some physical knowledge of the expected failure modes, you can predict (or assume) an acceleration factor for operating the part at high temperature. For example you might find that some type of failure doubles its occurrence (halves its MTBF) for each 10 degrees of increased temperature. In that case, you could simulate 10 years of operation in 3 months by operating at 55 degrees above the design limit. 

However, good estimates of the acceleration factor depend on good knowledge of the important failure mechanisms and are best if validated by long experience with the particular types of parts; so its best to get this information from your vendor. Also, there's a limit to how much acceleration factor can be used --- at some point you could accelerate some secondary failure mechanism so that the failures you see in the test don't represent the type of failures you see in the field.