As far as my experience gets, this can happen because:
- Punctures in the tube are too tiny.
- Valve is dirty, damaged or the seal is too old/has been overheated.
- The tube is old or has been chemically damaged, so has been turned more porous than normal.
For tiny punctures, inflate the tube a lot more when testing and be more patient when testing, i.e. allow a little more time each time you sink the tube in water, and watch for bubbles that develop slowly and remain stuck to the tube. Don't be afraid to over inflate the tube, it won't blow up, but if it does, it was already too old or chemically damaged. You can safely inflate a tube up to three times it's normal "non stretched size".
If you ride without valve caps, dirt can get inside the valve or even the tube, later returning to the valve and getting stuck between the valve's sealing surfaces. This is hard to test without inflating the whole tire assembly and pumping it to maximum pressure. The valve can sometimes be cleaned by inflating to max pressure and deflating quickly a few times (2-4) IF this is attempted but does not solve the problem, look for a way of recycling the rubber!
Seals in bicycle tubes are usually made of very soft rubber, which looses flexibility with age, even in the shelves, so, if you ever buy a tube that leaks air by its valve, try to get it exchanged for a different brand or one that is from a different/newer batch. Schrader and Dunlop type vales cores that are easily replaceable and with luck can be cheaply available. (In my country I can get 3 to 5 Dunlop cores for less than US$ 1.00) Schrader Valves are the same used in car's tires so maybe you get used ones for free at tire shops. (AS for presta valves, I do not know of replaceable cores...).
I have had ruined dunlop valve cores due to rim brake overheating. As mentioned before, rubber seals are really soft and can be easily damaged. They become dry and cracked with age and heat. To avoid heating of rim brakes, try to keep your rims and pads clean and free of oil and crystallized debris and observe a good braking technique in long steep descents. Schrader and presta valves are much less prone to this kind of problem, however, rim brake heating can also damage the valve neck (the union between the tube and the valve stem) weakening the union, producing leaks that are worthless to repair and sometimes very difficult to detect with the usual sink-in-water test.
As for old or degraded tubes, it's fairly easy to detect the problem. Inflate the tube so it becomes stretched. The tube should stretch in a more or less uniform fashion, and keep almost the same texture all-round. A very old tube degrades naturally, but chemicals can also damage rubber, accelerating its aging. Be careful if you store, repair, or use your bike, tires or tubes in places with existence of powerful solvents, paints with very volatile properties and the such. If you ever suspect that a tube has been chemically damaged, carefully inspect the tire as well.
A tube that has aged out will stretch more in its weak parts. These parts would present a less smooth surface, showing cracks and pores which appear not to be leaky. Those parts of the tube won't have the subtle shine of a clean rubber tube. Another sign of a weak or aged out tube will be the appearance of inexplicable punctures, such as tiny perfectly round holes in the part of the tube that faces a good clean rim tape. These fine holes appear by themselves, not by any object damaging the tire or tube.
Finally, all that I have stated, comes from my experience, and I must say, that it varies somewhat due to quality, design, raw material and manufacturing processes used by various brands.