Yes; your view would be particularly apt if the angel gave any indication of time, e.g. "Behold thou shalt conceive in thy womb this evening".
There is this historical tidbit though: betrothal was lengthy but it often involved sharing the household with the future wife. The wedding would be a huge feast that would involve many relatives, some coming from far away, and it would consecrate the betrothal, while at the same time requiring the household economy to be already in place at least in the kitchen, if not in the bedroom. Pregnancies would on occasion result form such betrothals; they were frowned upon and tended to accelerate the date of the wedding, but they were not unheard of. This is consistent with the Gospel of Matthew that calls Joseph Mary's husband and indicates that he was thinking of "putting her away" (Mt 1:19), that is, send her out of his house. Thus even if Mary took the angel to mean "you will conceive soon", it would still be a possibilty given that she shared the house with Joseph.
Do you have a reference for this? I only ask as I have never heard of it, especially as the Jews of Jesus' time were quite circumspect.
My understanding is also that betrothals were lengthy, but I had never heard of the sharing of a household before. Admittedly, that would help with that pesky verse 19 :) as I don't think they were married by verse 19.
In any event, I just don't see how Mary could have possibly received the news that she was going to give birth to the Savior, and it would be a result of a union outside of marriage. Since it was frowned upon, how could this be a fitting entry into the world for our Lord? Mary must have understood that.
But one thing we do know is that Mary WAS pregnant soon after the angel's visit. At the absolute maximum, it was three months later. We know by correlating the pregnancy of Elizabeth. The angel tells Mary that her cousin is six months along, and then John jumps for Jesus in the womb. That's a small window.