The real answers may depend on the specific kind of device, and on the type of battery it's supposed to use.
A common problem is corrosion on the contacts where the battery goes in. When you insert a new battery, you may scrape through the corrosion and make enough contact to look like it's on, but the resistance due to the corrosion may not allow sufficient current for the device to function.
Some devices will "turn on" if the batteries are installed backward - with the wrong polarity, but won't function. We'll assume you were careful about that.
Some devices are very sensitive to specific battery type. If it's designed to use Alkaline batteries, for example, it may not work on cheap Carbon Zinc batteries. If it's intended to use rechargeable Lithiums, Mercury rechargeables won't do.
If you're using a "battery tester" to determine that the battery has "like 90%" left, the tester must usually be specific for the kind of battery you're testing. A mercury rechargeable has a lower voltage than a carbon zinc, so the voltage - the only thing a tester usually checks - that indicates good charge on a mercury may be close to dead for another kind of battery.
Or there may have been a nuclear incident near your neighborhood and the EMP (Electromagnetic Pulse) effect may have burned out an internal component in the devices that are acting strangely. More likely a failed or "weak" diode or capacitor inside a device may give behaviour similar to your description, although it would be unusual to find a "clump" of devices all with the same kind of failure.