Some websites (such as OLTAs) don't reflect real time inventory. They only check for current fare class availability when you proceed to purchase a ticket. This can sometimes explain the messages saying that flights are unavailable or that the price has increased.

Major airline websites tend to query inventory in real time, but often cannot book flights on other airlines, including partners in many cases, although a real live human travel agent can combine specific segments together on a single ticket/itinerary in many cases. On the other hand, some combinations of carriers don't interline and LCC/ULCC flights generally must be ticketed separately.

Also, some travel agents are in the business of selling award tickets, often for long haul business class itineraries, using other people's miles (purchased or stolen). This violates airline rules and can strand you abroad. The MO seems to be advertising a price without details about airline/routing/schedule or requiring specific personal inquiries with these people often not checking for award inventory until you agree to buy the ticket from them.