If a mother becomes pregnant while nursing, her milk supply will become turbid and (unless an alternative is found) her nursing child may die (Yevamot 42:)
סתם מעוברת למניקה קיימא דלמא איעברה ומעכר חלבה וקטלה ליה
So how good a contraceptive is nursing? In a word (well, actually two words) it depends. In the first 3-6 months after birth, and if the baby is fed only on breast milk, some claim that breast feeding is a pretty good contraceptive, and is effective about 98% of time. But if mum skips a feed here or there, or if mum's periods have re-started, all bets are off. Here's data from an old paper on the topic. Take a close look at the last column- the failure rates per 100 women.
Those are high failure rates -as high as one in five - which makes it a pretty unreliable contraceptive. A review of breastfeeding as a contraceptive was published in 2003 in the widely respected Cochrane Reviews; it concluded that "[f]ully breastfeeding women who remain amenorrheic have a very small risk of becoming pregnant in the first 6 months after delivery when relying on lactational sub fertility". However, - and this is really important - it is not possible to know when amenorrhea is likely to end, and so an IUD is suggested as additional contraception wherever possible.
Overall, the talmudic suggestion that conception is possible while a mother is breastfeeding her child is, scientifically speaking, spot on.