In many TiVo models, when a power supply goes bad, one or more capacitors show signs of failure. A failed capacitor will bulge at the top and might have discoloration there as well. If you have burst capacitors, you certainly have a bad power supply. We have a replacement TiVo power supply page. If your capacitors look fine, you could still have a bad power supply. The only way to know at that point is to swap out the part with a known good one and see if the problems are affected.
Notes on specific models:
TiVo Series2 models starting with 240 service numbers: The power supply in this model fails pretty often these days due to the age of the unit, but this model rarely shows bursting capacitors.
TiVo Series3 models (all): These are the most likely TiVos to have failing power supplies. If you haven't changed your power supply by now, you likely need to. The vast, vast majority of the power supplies we sell are for these models. See the pics below. We also have videos showing how to replace these.
TiVo Premiere Series4: The early version of the Premiere line (746 and 748 models) have power supplies that fail decently often. The later models (750 and 758) rarely fail.
TiVo Series3 Failed Power Supply
TiVo HD Failed Power Supply (not all TiVo HD models have a power supply that looks like this)
TiVo Premiere Series4 Failed Power Supply
See failed TiVo power supplies from older models below
In the colored portions of the pictures below, you'll see capacitors that are bad. Look at the shape of the highlighted capacitors in comparison to the others. The colored capacitors have tops that are domed and, in the case of the last picture, have discoloration at the top.