Yes. A woman can get pregnant any time of the month. Use protection every single time you have sex, no matter what time of the month it is. This is to protect you from STD's and pregnancy.
The story or the mechanism of conceiving a baby is this:
Imagine the shape of an alphabet Y which has two arms left and right joining to come down on a single leg. Now the left and right arms (called fallopian tube) have their mouth opened against a bunch of grapes on either side and these grapes (are ovaries) which take turn one at a time on one side, and timely ripen one by one a month and fall into any one side into the arm's mouth and roll down the arm to the meeting junction called uterus and finally exit via the leg called vagina. A penis can some time ejaculate as far as the exact junction (uterus) and if around that time an ovary has reached after rolling down the left or right arm, pregnancy is effected by the sperms attacking the ovum, one of the millions of sperms ejaculated wins the race among them and pierce the ovum, and the ovum is fused with sperm or fertilized to produce an embryo, the ovum's wall hardened to protect from any further attack of any sperms. (So one minus one million is a waste ha ha ha )
Now the ovum attaches to the wall of the uterus to build a concrete connection of blood vessels that eventually is the umbilical chord.
Now the point here is that if one day before your period means that the OVUM would have come as close as the vagina and its therefore can easily meet (its boyfriend ) sperm and get fertilized.
Hope I made it thorough.