The answer to this question relies largely on context. I can only speak on honey bees, as they are my model species, but at the individual level it is rare for a forager to collect both honey and nectar simultaneously. It is typical for a forager to specialize on foraging on one food substrate or the other. This is optimal for foraging efficiency. However, at the colony level (the reproductive unit of eusocial bees) many foragers may be collecting nectar while others are simultaneously collecting pollen. This dynamic changes depending on the needs of the hive and is especially dependent on seasonality. Honey bees primarily rely on pollen for feeding brood, which are not generally produced all year, so there is a hard cap on the amount of pollen a colony will store until the tradeoff of producing new comb and other costs outweigh the utility. Juxtaposition this with nectar, which when stored becomes honey. This is the predominant food resource that the honey bees will rely on all year, especially in winter or nectar shortages. A colony will generally continue to gather nectar until the storage comb is completely filled.

In honey bees : both behaviors are possible : at individual level, it depends of the individuals and of the type of flower also. You can observe in a same flower the different type of behaviours depending on the hour of the day.

For instance, i refer to observations that i did on apple flowers during one full month : in the morning, you'll see a lot of pollen specialized honeybees collecting pollen exclusiveley. Later in the day, you may find more nectar or nectar+pollen foragers. Some flowers are so plenty of pollen (like dandelion for instance) that the honeybees cannot avoid to brush themselves to collect the pollen they have in all their body, even if they are mainly driven by nectar interest. In this case, the pollen is deposited as soon as possible in the hive, leading to full pollen frames that can even block the queen egg laying. In most of the cases, the honeybee have to develop a specific behavior to collect nectar or pollen, and they "choose" the right behavior in function of different factors.

In summary = there are many cases.

For wild bees and other pollinators, the behaviors may be even more complex.

There is huge difference across the species, especially based on their body size and flight energy. However most bees typically collect either pollen or nectar on foraging trips, although some can carry both simultaneously.