Two different things get called "won't render," and only one of them fixes itself with more time on the heat. Fat that's still pale, semi-solid, and cloudy is just running behind — the structure holding it hasn't given up yet, and it will if you leave the pan alone a while longer. Fat that already smells sharp or scorched is finished, and no amount of extra time brings it back.
The fat and the crisp are two separate races
Rendering means breaking down the connective tissue that holds fat inside its cells, and that takes time at a gentle heat no matter what the fat is attached to. Crisping the surface above it is a different process entirely — moisture has to leave the surface and the exposed proteins have to brown, which happens fastest with dry, fairly high heat. Both races start when the pan does, but they don't finish at the same time, and pushing the heat up to speed one along usually wins the wrong one. Turn the burner up and the surface dehydrates and browns while the fat underneath is still locked in place. That's how a duck breast ends up looking done on top while the layer beneath the skin is still mostly solid.
Cloudy, greasy fat that just needs more time
If nothing smells off, cloudy or greasy fat isn't a failed batch — it's an unfinished one. The milkiness is water and bits of protein still suspended in the fat rather than separated out of it, and that only clears once the connective tissue around them has fully broken down. Lower the heat rather than raising it, and keep going until the sound in the pan drops to almost nothing and the liquid turns from cloudy to clear. There's no raw-meat safety issue hiding in this — it's a texture and shelf-life question, since fat left cloudy with trace water in it goes off faster in storage than fat rendered all the way through.
Skin that renders but stays leathery
This is the opposite mismatch: the fat comes free, but the skin above it never crisps. Usually the pan was crowded, so trapped steam kept the surface from ever drying out, or the skin got pulled the moment the fat stopped sizzling instead of being left a little longer to actually brown. Give it space in the pan, press it flat so it isn't cupping and holding liquid fat against itself, and finish with a short stretch of direct, dry heat once the fat itself has mostly rendered out.
Cracklings burnt while the center is still pale
This happens when a piece browns from the outside faster than heat reaches its middle — usually because it was cut too thick, or because one edge of the pan was running hotter than the rest. The burnt exterior isn't a sign the interior finished; it's a sign the interior never had the chance. Smaller, more even pieces close that gap. If you catch a few overcooked pieces early, pull them out before they sit in the fat and taste the fat on its own — burnt bits removed quickly rarely ruin the batch.
When the whole pot is done for
Fat is an efficient solvent for whatever's dissolved in it, and burnt compounds are no exception. Once the smell of the fat itself has gone from clean and savory to sharp and acrid, that flavor has spread through the whole pot, not just the blackened pieces sitting in it. Straining won't pull it back out. At that point the batch is for the trash, not the fridge — start the next one with smaller pieces and a lower flame.