Can I visit India on my business visa without doing any business?

I do this all the time.

I am also a US citizen and with a 5-year visa. I have entered India many times on that visa or a similar one. No one ever inquired if this trip in particular was for business purposes, nor would I know how I would have proven it. I don't know if I was in technical violation of the visa, but I do know that no one took an interest.

Is it legal?

Strictly speaking, no. Just like it's illegal to do business on a tourist visa, it's also not allowed to use a business visa for a pure tourist trip.

A change of purpose is not permitted once you reach India. Depending on the nature of your journey, apply for the correct visa category. ^Source

A Business Visa is strictly given to those who would like to make a business related trip to India such as making sales or establishing contact on behalf of a company outside of India. ^Source

Here's a link to an official document listing permitted purposes for a business visa. Tourism is not one of them.

Will/can/may you get away with it?

Yes/probably. It's certainly possible (or even likely) that you'll get away with it. I know westerners who have used business visas for years, without doing much (or any) business at all. (You have probably also noticed @Malvolio's answer, which recounts similar experiences.)

The main problem is that you're going to have to lie to the immigration officer at the airport. He/she is likely to ask you "What is the purpose of your visit"? You will have to lie and say "business". It is not at all unlikely that the officer will keep enquiring and say "What sort of business"? "With which company?", etc. Can you lie in the face of such questions, and keep a straight face? Because if you tell the truth and say "tourism" you will most likely be denied entry. So keep in mind that your plan implies lying to the immigration officer. If you're caught doing this you may also be blacklisted from receiving visas in the future. Sure, you're unlikely to be caught, but it's not impossible.

It's also possible that you'll be questioned upon leaving India. I've been questioned at least twice while leaving India. One time, they even asked me where I had been staying, what I had been doing, etc. (The reason for this was that my passport was damaged, but anyway.) The other time it was simply some quick questions from the woman giving me the exit stamp. Again, you'll need to be prepared to lie.

It's very unlikely that you'll have any problems during the stay itself, unless you have some bad luck and somehow get in trouble with the police or something like that. Indian police and officials do like to enquire about details if someone gets their attention. Again, it's extremely unlikely but not completely impossible.

My advice

I don't recommend lying to border officials. Yes, you're likely to pull it off, but it just adds extra stress to the whole trip. It can be gnawing at you in the back of your head, and ruin what is supposed to be an enjoyable holiday. Also, it's a low risk, high impact decision. Let's say 99% of the time you make it. Well, the 1% of the time that something bad happens, it can have serious consequences. you can be deported, denied entry or even blacklisted (which may include other SAARC countries as well.) So, my advice is: Just get a tourist visa (or visa on arrival, eVisa, or whatever is convenient.) Yes, it may mean your business visa will be cancelled, but if you are making repeated trips for business and pleasure to the other side of the world then presumably you can afford a couple of application fees and doing some paperwork.

That said, it's obviously your decision. So weigh it over carefully and do what you feel is best, after having taken into consideration what you've read here.

One option I've been asked about is that you could plan a few business activities during the trip. It can just be a few meetings or something. (Presumably you already have business connections in India.) You're then no longer lying if you say "business." You'll also have something to refer to (without lying) if they ask you "which company", etc. In any case, arriving in a business suit will probably help you. Attire is very important in India.

You don't mention what exactly the purpose is. I'm assuming it's tourism. However, if it is, for example, surrogacy then don't do it on anything other than a surrogacy visa. I have no reason to assume that surrogacy is your intention, but I'm mentioning it because you specifically don't mention tourism or say what the "other purpose" is.

^{(And actually, surrogacy is illegal now, so you'd only be able to do that anyway if it was commissioned before the ban.)}