iOS: Bluetooth LE background app
It is indeed challenging to get that functionality to work reliable, since iOS has a very conservative power saving mechanism and developers are not making things up.

The problem is: While there actually is a way for Apps to use Bluetooth LE while in the Background, it's very limitated. Typically, to connect to a BLE device like Noke, the App needs to scan on for the device which advertises is presence. Only after a scan is received, the app can actually connect to the device. A device issues a scan using iOS system services and registers a callback, which means the OS "calls"/"activates" the app if a device that was scanned for is found.

When scanning for a device, there are several parameters that can be tuned that affect power intensity but also latency when interacting with BLE. In gerneal, you need to balance these parameters for the best experience. In foreground, scanning for and connecting to devices is very quick on iOS, but in background, iOS ignores some options developers can configure but foremost, it reduces the frequency of the scan which greatly reduces the power usage, but also introduces a lot of latency. iOS also decides when to issue the callback to the app described above.

I made the discovery that power saving on iOS 11.1 is way more aggressive then on iOS 10. Are or where you using iOS 11 when those problems occured?

Source: https://developer.apple.com/library/content/documentation/NetworkingInternetWeb/Conceptual/CoreBluetooth_concepts/CoreBluetoothBackgroundProcessingForIOSApps/PerformingTasksWhileYourAppIsInTheBackground.html#//apple_ref/doc/uid/TP40013257-CH7-SW1