It is possible, but can often be avoided with proper care.
Waxing removes the entire hair, root and all, from the follicle. Your skin naturally hardens and thickens over the area that was waxed so it is important to keep that area gently exfoliated. Use a gentle scrub in that area about 2 days after waxing and continue to exfoliate 1 to 2 times a week thereafter.
The bikini, lip, and underarms are where we tend to perspire the most. Perspiration and moisture cause bacteria to grow. Once that area has been waxed, the follicle is wide open to this bacteria creeping in.
Try to wear loose-fitting clothes for a few days after to give your skin a chance to breathe and reduce moisture.
You can also use a mixture of tea tree and essential oils. These are antimicrobial and antibacterial and will prevent irritation.
Try to avoid heavy exercise that will increase perspiration and chaffing for about 24-48 hours after waxing.