Is she avoiding me or am I insecure?

There is this girl I know online. I've known her for roughly seven months. I didn't talk to her much when I first met her, but after a couple months I started to talk to her more and more.

Since I've started talking to her more though. I've noticed something. Sometimes she is really hard to get a hold of. I only know her online, but I have various ways of contacting her.

Sometimes when I try to talk to her on skype, she won't answer. She only replies to a few of my text messages and often when I am talking to her. She will stop talking for no reason.

I was starting to think she was trying to avoid me, but when I asked her about it. She said "I love you too much to avoid you", and not only that. Whenever I ask her why she wasn't answering me at some point. She always has a reason. Like it was her brother using her computer or something along those lines.

I want to believe her, but there are times where I am almost 100% certain she is there. She still won't answer though. I've given her chances to come clean and admit it, but she always denies it. No matter how unlikely her excuse is.

She has no reason to go through this much effort to avoid me, but still pretend to be my friend. She doesn't have anything to gain by pretending to be my friend, but some of this is just too weird and I am very confused.