Quoted from G-P-E:

Tantalum caps are wonderful caps but don't like two things.
Over voltage.
Reverse voltage.
The fact that it is blowing can be indicative of another problem.
What is the voltage going into cap? And what is the voltage rating of the cap?

Quoted from G-P-E:

I'll have to call BS on them not aging well. We have systems pushing 40 year mark with tantalum's. If it wasn't for routine maintenance, the electrolytics would ALL be shot many times over (high ESR and/or low value due to being dried out) but the tantalum's will still be within spec. We routinely bring some of the equipment back every 8 years or so for revalidation. We now have the techs just replace all the electroltyics. In that time, we have replaced *one* tantalum (and that was due to operator head-space). Worst is when the electroltyics spew out electrolyte out the bottom and nobody notices until the damage is done.
What does cause them to age prematurely is what I listed -- they do not take kindly to over voltage or reverse voltage. In a properly designed circuit, tantalum failures are rare.
On the old board like shown in the video - we have more problems with the old carbon composition resistors than tantalums. They absorb moister over time and change value. Did you notice there were *no* electroltyic capacitors on the board shown in the video? Why do you suppose that is the case with this board? If there were, they would have ALL been replaced at least three times before the one tantalum was replaced.
And since you were speaking of power up stuff not powered on in years. It's typically not the tantalum's that are the problem (and party poppers), it's the electrolytics. For this we created a custom 'capacitor reforming' cage at work. Even bringing up the electroltyic caps slowly will cause some of them to blow... even if the caps have never been used. We use the cap reformer for unused electroltyics that have been in storage for up to about 10. Once they hit ten, in the trash they go.