Joolie
You are right with what you are saying it is possible..
To really have a idea if this happening to you...
Is to take your BG reading when you go to bed, and then one when you get up, and compare these results... These should be roughly the same..
If not, then over a couple of nights, take a reading durning the night, 3am is considered to be the point when hormonly the body is at it's lowest ebb, so start with this one, the a 2am, 4am etc.. to see if you can see any patterns.
Another tell tell sign, that might suggest that you've sufferes a mild hypo while sleeping is when you wake up, if feels like you've got a hang over!!
It could also be that you are suffering with Dawn Phenomenon, this is were the body starts it wake up process, where it increases the level and amount of hormones, along side increaseing the amount of glucose being delivered into your system, from your liver..
To counter act this effect, is to split your background insulin into two injection to am/pm then you can adjust the dose to suit the needs of the morning and/or the night time needs to pervent and stablise your levels.