Let's suppose I have a test which gives me the dosage of an analytic in the blood. The results of the assay are in a range of 0 and 1000; all subjects who have a value higher than 1000 will be categorized as ">1000".
The question is: how can I manage this variable since a significant proportion of the cohort will have levels >1000? how can I fit a regression in which the result of the assay is the dependent variable?
I'd guess that implicit in your question is not just "what can I do?" but also "what should I do?" The first part is relatively easy to answer, in part because there is so much you can do! The second part will require you to determine what makes the most sense given your setting.
To make sure I understand your question, suppose $Y^*_i$ is the dosage of the analytic in individual $i$, and $X_i$ is some other variable. You observe data $(Y_i,X_i)_i$ for $i=1,\dots,n$, where $Y_i = \min(Y^*_i,1000)$. To make sure we are on the same page regarding terminology, you have a censoring problem. In particular, this is not a truncation problem, which is often confused with censoring (see this question for more on that)
You seem to be interested in a regression model where your dependent variable is censored, as is the case when it's $Y_i$.
1. What you should not do
Before discussing some options, I'd strongly caution against the approach suggested in the first comments of your post, which entails just assigning random values to those with $Y_i = 1000$. To illustrate intuitively why this is problematic, suppose $X_i$ is gender, and suppose all males have $Y_i = 1000$ (i.e. $Y_i^* > 1000$ for all males) and all females have $Y_i < 1000$ (i.e. $Y_i^* < 1000$ for all females). If you just randomly impute $Y_i^* > 1000$ (male data) using $Y_i^* < 1000$ (female data) and run a regression, you will find no effect of gender, when clearly there is one! You may wonder: "Well, what if I impute conditional on $X_i$?" Again, the same problem persists: suppose that males with $Y^*_i < 1000$ have $Y^*_i$ that are like females, but half of males have $Y^*_i > 1000$ whereas all female have $Y^*_i < 1000$. Again, there's a clear difference between males and females, but if you impute all males above $1000$ to have $Y^*_i$ like those males below $1000$, you will erroneously conclude there is no difference between genders, when again you'd expect one!
The lesson here is that a censoring problem should not be solved by imputation, unless you have a very good reason to believe it can.
2. What you may want to do
As you may have guessed based on the above, what you should do will depend heavily on your setup and what you think is reasonable. As a first step, you can always only look at $(Y_i,X_i)$ for $Y_i \leq 1000$. A regression with this data should be fine, except you have to be very clear about the interpretation! Even if the coefficient is the effect of $X_i$ on $Y_i$, here it is that effect for those with $Y^*_i \leq 1000$. Whether that matters depends on your setting. Given that many have $Y^*_i > 1000$, maybe this is not an interesting question.
A general class of regression models such models are called Tobit models, and I'd suggest you check out that page. For more information, check out this note on censored models. These Tobit models typically assume that $$Y^*_i = \beta X_i + \epsilon,$$ where $\epsilon \perp \!\!\! \perp X$ and $\epsilon \sim N(0,1)$. This normality assumption can also be relaxed: see the above note for relaxing that issue using Powell's CLAD Estimator.
Another area you may wish to explore is that of survival analysis, where this issue is very prevalent.
