In this work, we combine the two notions of timely delivery of information to study their interplay; namely, deadline-constrained packet delivery due to latency constraints and freshness of information. More specifically, we consider a two-user multiple access setup with random-access, in which user 1 is a wireless device with a queue and has external bursty traffic which is deadline-constrained, while user 2 monitors a sensor and transmits status updates to the destination. We provide analytical expressions for the throughput and drop probability of user 1, and an analytical expression for the average Age of Information (AoI) of user 2 monitoring the sensor. The relations reveal that there is a trade-off between the average AoI of user 2 and the drop rate of user 1: the lower the average AoI, the higher the drop rate, and vice versa. Simulations corroborate the validity of our theoretical results.