Don't know if you are actually "pressing" on the screen or that was just word choice.
But, my, uhm... someone else that lives with me who shall remain nameless...
Had to be re-trained on using a touchscreen. The key word here is "touch". Maybe it was too many years spent as a typist, I dunno. But, she had a real problem using a "touch" screen and wanted to "press" everything. There is a difference between touching and pressing. She's experienced this problem across multiple devices from the original Kindle, Fire, Nexus and iPad with them occasionally not recognizing a "press" on the screen. Every time we get a new device she goes back to pressing instead of touching.
Never did try to figure out if it was just random or only certain areas of the screen that didn't respond to a press or not and took me awhile watching her have problems where I could take the same device and have no problem to figure out what was happening.
Not your scenario? Try opening the Notes app and typing there. I would assume it's possible there is something wrong with the device itself, so best to find out now and exchange it for another if so.