Hello all, I recently bought the Lenovo 7inch tab. I had the same problem of the Lenovo not turning on. I got a technician at my company have a look at it and he realised that most Lenovo tablets don't charge from custom USB ports that you can find in the style of Multi USB hubs or computer/laptop ports.
The solution: connect the lenovo directly to a single USB wall outlet.
This worked for me. Initially, I thought it was a LCD issue or a motherboard problem but, all it was that it wasn't being charged correctly. I was using a multi USB port to charge it and all it did was hold the power, not charge it further. Hope this helps :)