A 'typical' house in the US is built to carry a 'live load' of 15 to 20 pounds per square foot. A live load is defined as anything that exists in the space except the things directly attached. The floor boards and carpet are not part of the live load but people, furniture, fish tanks, pianos and so forth are. When you think about the fish tank that weighs about 10 pounds per gallon on average, none of them can be supported by just the square footage on which they rest. Instead you go back to the definition and find that the rating is based on a load evenly distributed over the floor. A 10 by 12 foot room with a 15 pounds per square foot live load rating can theoretically hold a total of 15x12x10 or 1800 pounds. That value is only correct if the load is even distributed though. It would give a rough idea of the total weight that could be placed in the room without damage, which would be about 180 gallons. If you were to start lining up fish tanks in the room, it would quickly exceed that weight limit, so yes, there is a definite limit to what can be placed in a room. An even more important factor is the location of the tanks. A single 180 gallon tank in the middle of the room would probably be in danger of ending up at least one floor down fairly quickly. The weight that a floor can hold depends more on where that weight is located than how much it is. In the middle of the span of a floor joist, the weight that can be supported is at its minimum. Nearer the ends of a joist, it is higher. If you can spread the weight among several joists, the maximum weight without any damage is greater.
The original poster was talking about placing a tank parallel to the joists so the weight would only be carried by one or two of them which is riskier than running the tank in the other direction. If the space under the floor is accessible, it would be best to brace the floor joists that will be carrying the weight directly under where the tank will be placed. That way, the weight could be transmitted to whatever stable structure exists below the joist rather than being taken up by the joist itself.